RCCC UPDATE

Residual Curvature Cyclic Cosmology (RCCC) Rhythm October 17, 2025 Abstract We introduce Residual Curvature Cyclic Cosmology (RCCC), a novel framework in which a localized, covariantly conserved geometric imprint from a prior cosmological epoch a spacetime scar described by a residual curvature tensor ∆µν persistently modifies large-scale dynamics and can drive successive cosmic cycles. Unlike matter-based bounce scenarios or global conformal identifications, RCCC places the memory of a previous aeon in an effectively localized geometric deformation that (i) contributes an extra term on the geometric side of Einstein’s equations, Gµν +∆µν = 8πGTµν +Λgµν, (ii) sources late-time departures from standard ΛCDM, and (iii) can produce a classical turning point (expansion contraction) without exotic NEC-violating matter. We show how a scar whose averaged density scales as ρs(a)∝a−ncan mimic a time-varying dark energy component; if n < 0the scar effectively activates and grows with scale factor, producing dynamical signatures including apparent phantom-like weff(a)and a finite future root of H2(a)=0. We present a concrete microphysical toy model (a localized scalar condensate) that yields an explicit ∆µν, derive the exact turning condition for the background, and quantify entropy bookkeeping: the scar can sequester fine-grained information into a scar Hilbert space of capacity Smax ∼πR2 H/ℓ2 P, allowing the exterior to re-nucleate in a low coarse-grained entropy state. RCCC produces sharpened observational predictions directional large-angle CMB anomalies, distinctive ISWLSS cross-correlations, and a specific weff (z)evolution and a suite of falsifiable inequalities (e.g. scaling exponent n= 2, entropy ratio bounds). We provide complete analytic derivations, numerical recipes (CLASS/CAMB patches), LTB collapse realizations, semiclassical bounce estimates, and data-analysis pipelines to confront RCCC with present and forthcoming observations. 1 Conceptual foundations This section lays out the logical and mathematical architecture that motivates the Residual Curvature Cyclic Cosmology (RCCC). We proceed in steps: (i) restate the problem in the standard GR + Λframework, (ii) introduce the scar hypothesis at the level of the field equations, (iii) derive the background cosmological consequences and the turning condition, (iv) discuss entropy bookkeeping and the scar as a geometric memory, and (v) outline observational mappings that make the hypothesis falsifiable. 1.1 Problem statement in GR + Λ Standard cosmology is governed by Einstein’s equations with a cosmological constant, Gµν + Λgµν = 8πG Tµν.(1) 1 Under classical evolution this system generically leads to initial curvature singularities (Big Bang) and for closed models to successive cycles only at the expense of monotonic entropy growth (Tolman’s argument). Attempts to resolve singularities or produce non-singular bounces typically introduce either exotic matter (NEC violation), higherdimensional dynamics, or quantum gravitational modifications. RCCC proposes a different route: a geometric relic of prior high-curvature dynamics persists as a tensorial deformation of the spacetime geometry that is neither a conventional matter field nor an ad hoc external forcing term. 1.2 The scar hypothesis: field-equation level We elevate the scar to a covariantly defined, phenomenological contribution ∆µν that modifies the geometric side of the field equations: Gµν + ∆µν + Λgµν = 8πG Tµν.(2) Key operational assumptions (to be justified or relaxed in microscopic realizations): 1. ∆µν is covariantly conserved, ∇µ∆µν = 0, ensuring Bianchi consistency. 2. ∆µν is spatially localized (or strongly peaked) around a comoving center xswith a characteristic comoving width σand is persistent on cosmological timescales. 3. On cosmological scales one can volume-average the localized scar to obtain an effective homogeneous contribution characterized by an averaged energy density parameter Ωs,0and an effective scaling exponent n, ρs(a) = ρs,0a−n,Ωs,0≡8πGρs,0 3H2 0 . 1.3 Background evolution and turning condition In a spatially homogeneous FLRW background with radiation, matter, cosmological constant and the averaged scar term, the normalized Friedmann function is E2(a)≡H2(a) H2 0 = Ωr,0a−4+ Ωm,0a−3+ ΩΛ,0+ Ωk,0a−2+ Ωs,0a−n.(3) A classical turning point (expansion →contraction) occurs if there exists at>0such that E2(at) = 0.(4) The algebraic turning condition isolates the scar contribution: Ωs,0a−n t=−Ωr,0a−4 t+ Ωm,0a−3 t+ ΩΛ,0+ Ωk,0a−2 t.(5) Two observations follow immediately: •If Ωs,0<0and n < 0(i.e. an increasing scar energy density with expansion), the left-hand side grows in magnitude with aand can cross the finite right-hand side at some atproviding a novel mechanism by which a presently accelerating universe can classically turn without exotic matter. •The scale dependence of the turning term is a discriminant. Classical global curvature corresponds strictly to n= 2. If observational fits favour n= 2 (especially n < 0), a localized scar mechanism is indicated. 2 1.4 Local stability of the turning point Differentiate E2(a)and evaluate the sign controlling the local approach: A≡at dE2 da at =−4Ωr,0a−4 t−3Ωm,0a−3 t−nΩs,0a−n t−2Ωk,0a−2 t.(6) To leading order near atthe evolution obeys ˙ε2≃H2 0(Aε)with ε≡(a−at)/at. If A > 0 the linear term is real and a generic crossing into contraction occurs; if A < 0higher-order terms dominate and a tangential touch or bounce behavior may result. Thus A > 0is the operational signature of a robust turning into contraction. 1.5 The scar as geometric memory and entropy bookkeeping A core conceptual challenge for cyclic models is the Tolman entropy growth problem. RCCC addresses this by positing that the scar encodesquantum mechanicallya large Hilbert space Hs(“scar microstates”) that can store the fine-grained details of collapsing matter so that the accessible exterior degrees of freedom that seed the next expansion are of low coarse-grained entropy. Define: Smax ≡πR2 H ℓ2 P (holographic capacity associated to the scar/horizon scale), Sin ≡coarse entropy delivered into the core. A necessary operational condition for effective entropy reset is Sin ≲Smax,(7) so that scar microstates can accommodate the incoming information without forcing the exterior seed to be high entropy. More refined statements replace “≲” with numerically calibrated fractions (see Secs. 56 and Appendix B for detailed estimates). From a quantum information viewpoint, the global state at the quantum cutoff time tQcan be written (in a suitable basis) as |Ψ⟩tQ=X i ci|ψi⟩s⊗ |Φi⟩ext, and the reduced exterior state ρext = Trs|Ψ⟩⟨Ψ|can be engineered by the dynamics and the scar’s spectral properties to have low coarse-grained entropy even when |Ψ⟩is globally pure. Operational criteria (scrambling time tscr, spectral density Js(ω), and coupling strengths) determine how efficiently information is hidden versus transferred. 1.6 Observational signatures and falsifiable mappings The scar mechanism yields concrete, falsifiable predictions: 1. Background: A time-varying effective dark energy, ρeff DE(a) = ρΛ+ρs(a), weff(a) = −1−1 3 dln ρeff DE dln a, which for growing scar (n < 0) can produce apparent phantom-like evolution or a measurable rise in DE density. 3 2. Large-angle CMB: A localized scar generates low-ℓdirectional temperature anisotropy via direct SachsWolfe and late ISW contributions. Template morphology differs from Penrose CCC concentric rings and from homogeneous curvature signatures. 3. ISWLSS cross correlation: If the scar grows at late times, it produces an ISW signal correlated with LSS in a distinct angular pattern aligned with xs. 4. Empirical discriminants: Fit E2(a)including an extra term Ωs,0a−n. If best-fit n= 2, and especially n < 0, and low-ℓCMB/ISW data prefer a localized template aligned with LSS, RCCC gains observational support. Conversely, tight upper bounds on |Ωs,0|and non-detection of directional ISW impose strong constraints on the scenario. 1.7 Conclusion of the conceptual foundations RCCC relocates the principal conceptual burden of cyclic cosmology from a global, finetuned initial condition to a localized geometric object the scar whose microphysical properties and causal structure determine whether the next aeon begins in a low coarsegrained entropy macrostate. The construction is conservative at the level of geometry (no ad hoc exotic fluids are required), falsifiable (clear distinctive signatures), and amenable to microphysical uplift (toy scalar condensate, holographic storage models, or explicit quantum gravity completions). The remainder of the paper provides explicit microphysical realizations, exact turning algebra, detailed numerical recipes, and observational pipelines to confront the framework with data. 2 Field Equation Formalism and Scar Implementation 2.1 Philosophy and field-equation placement We adopt the minimal, phenomenological modification of Einstein’s equations Gµν + ∆µν := 8πGTµν + Λgµν,(3.1) where •(Gµν)is the Einstein tensor of the spacetime metric (gµν), •(Tµν)is the ordinary matter–radiation stress–energy, •(Λ) is the cosmological constant (allowing (Λ = Λ(t)) in general), •(∆µν(x)) is the residual curvature tensor (the scar) a localized, non-dissipating geometric deformation imprinted by the primordial singular event. Algebraically the choice to place (∆µν)on the left-hand side (LHS) or to move it to the right-hand side as an effective stress–energy, Gµν := 8πG(Tµν +T(scar) µν )+Λgµν, T(scar) µν ≡ − 1 8πG∆µν,(3.2) is a matter of interpretation only. We will freely use both viewpoints: a geometric deformation (LHS) or an effective localized stress–energy (RHS). 4 2.2 Covariant constraint: conservation and Bianchi identity The contracted Bianchi identity implies (∇µGµν = 0). Taking the covariant derivative of (3.1) yields the exact relation ∇µ∆µν := −∇µ8πGTµν + Λgµν.(3.3) Two physically natural simplifying choices are: 1. Conserved-scar, constant-(Λ): assume ordinary matter is covariantly conserved, (∇µTµν = 0), and (Λ) is constant. Then ∇µ∆µν = 0.(3.4) We adopt (3.4) as the principal working constraint for the scar: it is a covariantly conserved geometric relic that does not exchange energy–momentum with the ordinary matter sector at low curvature. 2. Interacting-scar / time-varying (Λ): allow (∇µ∆µν = 0) with a compensating exchange between (∆µν)and (Λ) or (Tµν). Such models are possible but more model-dependent; they will be discussed later. From now on we will take (3.4) unless explicitly noted. 2.3 Symmetry, coordinates, and ansatz The scar defines a preferred spacetime locus (xs)(the Big-Bang center). For analytic tractability and to capture the leading phenomenology we adopt spherical symmetry about (xs). Use comoving FLRW coordinates (t, r, θ, ϕ)for the background metric: ds2=−dt2+a2(t)dr2 1−kr2+r2dΩ2. We parametrize the scar tensor as a localized radial kernel multiplying a time-dependent tensor structure: ∆µν(t, r) = κf(r;rs)Sµν(t),(3.5) where •(κ)is a dimensionful amplitude parameter to be constrained observationally, •(f(r;rs)) is a radial kernel peaked at (r= 0) and decaying for (r≳rs)(characteristic scar comoving radius (rs)), •(Sµν(t)) encodes the tensor structure and time dependence (e.g. isotropic vs anisotropic components). Below we develop explicit, physically transparent specializations of (3.5). 5 2.4 Perfect-fluid effective form (local EFT viewpoint) It is convenient to decompose (∆µν)relative to the comoving 4-velocity (uµ= (1,0,0,0)). The most general decomposition is ∆µν =α(t, r)uµuν+β(t, r)hµν + 2u(µqν)+πµν,(3.6) with (hµν =gµν +uµuν),(qµ)the energy flux, and (πµν)the anisotropic stress. Spherical symmetry implies (qµ= 0) and (πµν)proportional to the radial traceless tensor. Two useful effective forms are: (A) Local perfect-fluid–like scar (useful to move (∆µν)to RHS): ∆µν ≃8πG(ρs+ps)uµuν+psgµν +πµνf(r;rs).(3.7) Here (ρs(t)) and (ps(t)) are the scar’s effective energy density and pressure profiles (the (f)-factor enforces localization). (B) Tracelike / cosmological-constant–like scar (LHS viewpoint): ∆µν =κf(r;rs)gµν,(3.8) which corresponds locally to an effective shift of the cosmological constant inside the scar region. (Note: conservation (3.4) constrains allowable (f(r;t)) choices.) 2.5 Covariant conservation continuity equation and scaling Under the perfect-fluid specialization (3.7) with (qµ= 0) and (∇µ∆µν = 0), project the (ν= 0) component in comoving coordinates to obtain the scar continuity equation. Integrate over the scar comoving domain (D)(fixed comoving volume containing the localized kernel) and define the volume average ⟨·⟩. The straightforward derivation (standard fluid projection) yields d dta3(t)⟨ρs(t)⟩+a3(t)3H(t)⟨ps(t)⟩:= 0.(3.9) If the scar kernel is comoving (i.e. its comoving support is fixed) and the local equation of state is barotropic (ps=wsρs), (3.9) integrates to the familiar scaling law ⟨ρs(t)⟩ ∝ a−3(1+ws)(t).(3.10) Remarks. •If the kernel occupies a fixed proper volume (Vs)instead, the averaging introduces additional factors and the global scaling can differ; the comoving-kernel assumption is the simplest and conservative choice. •The sign of (ρs)depends on sign conventions in (3.7)/(3.2): since (T(scar) 00 ≡ −∆00/(8πG)), a positive (∆00)corresponds to a negative effective scar energy density (ρs), which is the mechanism by which the scar can cancel positive contributions in the Friedmann equation (see Sec. 3.7 below). 6 2.6 Explicit radial kernels and regularization We must ensure the scar is localized and that curvature invariants remain finite at (r→0). Useful regularized kernels: •Gaussian kernel (regular, smooth): fG(r) = 1 (2π)3/2σ3exp −r2 2σ2.(3.11) Characteristic comoving width (σ). •Compact support (top-hat) kernel: fC(r) = 3 4πr3 s Θ(rs−r),(3.12) with sharp radius (rs)and (Θ) the Heaviside function. •Yukawa / screened power law (longer tails with cutoff): fY(r) = 1 4π e−r/λ r·greg(r),(3.13) where (greg(r)) regularizes the (r→0) behavior (for example (greg = (r2+r2 c)−1/2)). Regularization condition. For any chosen (f(r)) we require that curvature scalars computed from the full metric (gµν)corrected by (∆µν)remain finite everywhere: Kret ≡RµνρσRµνρσ := O(1) as r→0.(3.14) Impose short-distance cutoffs (rc≳αℓP)(with (ℓP)the Planck length) to safely regularize semiclassical calculations. 2.7 Modified Friedmann equation: volume average derivation Volume-average Einstein equations over a large comoving domain (D)that contains the scar core (but is large compared to scar radius) produce the effective Friedmann equation. Take the time-time component of (3.2), average, and define (ρs(t)≡ ⟨ρs(t)⟩). One obtains H2(t) = 8πG 3ρr+ρm+ρs+Λ 3−keff a2(t),(3.15) where: •(ρr),(ρm)are the ordinary radiation and matter densities (volume-averaged), •(ρs)is the volume-averaged effective scar density (note sign convention see remark below), •(keff)is an effective curvature parameter which includes both global spatial curvature and scar-induced curvature backreaction. 7 Sign remark: since (T(scar) 00 ≡ −∆00/(8πG)), a positive (∆00)yields (ρs<0). Thus apositive curvature deformation can act as negative effective energy density in the Friedmann equation this is the mathematical source of the turning mechanism. Inserting the scaling (3.10) for the scar, ρs(a) = ρs,0a−3(1+ws),(3.16) and writing standard scalings for radiation and matter, ρr(a) = ρr,0a−4, ρm(a) = ρm,0a−3, we can write the normalized Friedmann equation E2(a)≡H2(a) H2 0 = Ωr,0a−4+ Ωm,0a−3+ Ωs,0a−3(1+ws)+ ΩΛ,0+ Ωk,0a−2,(3.17) with the usual definitions (Ωi,0=ρi,0/ρcrit,0). The turning point (at)is defined by (E(at) = 0). 2.8 Turning condition and role of sign From (3.17) the necessary condition for a finite positive root (at>0) in a universe with (ΩΛ,0>0) and positive matter/radiation densities is that some term be negative. This can be realized in two ways: •Mechanism I (negative effective scar density): (Ωs,0<0) and (|Ωs,0|)grows relative to (ΩΛ,0)with scale factor (e.g. if (ws<−1/3) or if (ρs)decays slower than (ρΛ)). Then (Ωs,0a−3(1+ws))can cancel positive terms and yield (E2= 0). •Mechanism II (curvature backreaction): the scar generates an effective curvature/backreaction term (Ω(scar) k,0<0) (positive spatial curvature contribution in the usual sign convention), so the geometrical term (Ωk,0a−2)balances positive energy. In this route (ρs)need not be negative. Both are geometric the scar modifies (∆µν)and thereby the effective geometry. 2.9 Linearized / Newtonian limit and potential For many observational estimates it is useful to work in the Newtonian limit on subhorizon scales. Define the scar’s effective Newtonian density ρs,phys(x, t)≡ −∆00(x, t) 8πG ,(3.18) (note the minus sign because of the (T(scar))convention). In the quasistatic limit the Newtonian potential (Φs(x, t)) obeys the Poisson equation on the FRW background (neglecting expansion when evaluating local potential at recombination): ∇2Φs(x, t) := 4πGρs,phys(x, t).(3.19) 8 For a spherically symmetric, localized scar with total effective mass (Ms≡Rρs,physd3x) and at comoving radius (r)outside the scar core, the potential asymptotically behaves as Φs(r)≃ −GMs r(r≫rs).(3.20) Connection to CMB SachsWolfe: the large-angle temperature perturbation for photons passing through a static potential at last scattering is ∆T TSW ≃1 3Φs(rrec).(3.21) This gives the qualitative scaling used to bound (|Ms|)(hence (κ)) from observed (∆T/T ∼ 10−5)at low multipoles. 2.10 Linear perturbation response (metric perturbations) In linear scalar perturbation theory (Newtonian gauge) on the FLRW background ds2=−(1 + 2Ψ)dt2+a2(t)(1 −2Φ)γijdxidxj, the scar acts as an external source for (Φ) (with (Ψ = Φ) in absence of anisotropic stress). Linearizing Einstein’s equations with (∆µν)yields ∇2Φ−3HΦ′−3H2Φ := 4πGa2δρ +1 2a2δ∆00,(3.22) where (δ∆00)is the perturbation in the scar’s time-time component (localized). In Fourier space or via Green’s functions this produces the modifications to the primordial potentials and the ISW effect; explicit expressions for the multipoles (aℓm)follow from projecting (Φs)onto spherical harmonics (we use the standard projection integrals in Sec. 8). 2.11 Curvature invariants: sample computation and finiteness To guarantee the scar is a physically admissible geometric deformation we must check that curvature invariants remain finite after including (∆µν). Working perturbatively in the scar amplitude (κ), write gµν = ¯gµν +hµν, hµν =O(κ), and expand the Kretschmann scalar to leading order K≡RαβγδRαβγδ =¯ K+ 2 ¯ RαβγδδRαβγδ +O(κ2).(3.23) Because (f(r)) is regularized at short distances (e.g. Gaussian with width (σ≥rc)), (δRαβγδ)is finite everywhere and therefore (K)remains finite. Explicit verification in a given ansatz (say (fG)and choice (Sµν = diag(η0, η1, η1, η1))) reduces to straightforward algebra; we state the necessary requirement: Regularity condition: choose (f(r)) such that (limr→0r−nf(r)<∞)for every (n) appearing in numerator denominators of the curvature expansion; in practice a smooth Gaussian or compact-support kernel with (rc≳ℓP)suffices. 9 Technical Appendix Scar Cosmology Analysis •scar_enable : true/false 2. Compute ρs(η)inside background module In background.c (or the equivalent), compute 1rho_s(a) = rho_c0 * scar_omega_s0 * pow(a, -3*(1+ scar_w_s)); where rho_c0 = 3 (H0)2/(8 ∗pi ∗G)anda =scalefactor. 3. Add scar perturbation source in perturbation module In perturbations.c (function that builds source term for metric potentials), insert the following additional contribution to delta_rho (Fourier space): 1// pseudocode snippet inside perturbation source assembly (for each k and tau) 2if (scar_enable) { 3double k = k_array[index_k]; 4double sigma = scar_sigma_Mpc * h_over_Mpc; // convert if necessary 5double ftilde = exp(-0.5 * k*k * sigma*sigma); 6double rho_s = rho_c0 * scar_omega_s0 * pow(a, -3*(1+scar_w_s)); 7double delta_rho_s = - rho_s * ftilde; // negative sign per T_scar = -Delta/(8piG) 8delta_rho_total += delta_rho_s; 9} Then delta_rho_total is used as usual in the Poisson/constraint equations. No modification is needed to the radiation/matter continuity if scar does not exchange energy with them. 4. ISW & source functions The time derivative of the potential Φ′is automatically computed by CLASS once the modified Poisson equation is used. ISW contributions will appear in the computed CMB source functions. No separate change is required to the line-of-sight routine beyond the standard usage. 5. Initial conditions & small-k Because scar is a localized external source at large scales, it does not change early-time adiabatic initial conditions if |ρs(arec)| ≪ ρr(arec)|. For safety, enforce input constraint |rho_s(a_rec)| << rho_r(a_rec) to avoid spoiling initial conditions. 6. Output options Expose scar parameters in output summary and produce: •Cl_TT_scar.dat (TT including scar), • cross-correlation files for matter–CMB ISW. B.3 CLASS pseudocode block (ready to port) Paste this high-level pseudocode into perturbations.c near where delta_rho is formed: 1/* --- scar parameters (read from input) --- */ 2double scar_enable = params ->scar_enable; // 0/1 3double scar_omega_s0 = params ->scar_omega_s0; 4double scar_w_s = params ->scar_w_s; 5double scar_sigma = params ->scar_sigma_Mpc; // comoving Mpc 6 7/* inside loop over k and tau where you compute delta_rho_total */ 8if (scar_enable) { 9double k_phys = k; // CLASS uses k in 1/Mpc Technical Appendix Scar Cosmology Analysis 10 double sigma_phys = scar_sigma; // in Mpc 11 double ftilde = exp(-0.5 * k_phys * k_phys * sigma_phys * sigma_phys); 12 double rho_c0 = 3.0 * pow(cosmo ->H0,2) / (8.0 * M_PI * G); // H0 in 1/ Mpc units etc 13 double rho_s = rho_c0 * scar_omega_s0 * pow(a, -3.0*(1.0+scar_w_s)); 14 double delta_rho_s = - rho_s * ftilde; // effective scar delta_rho 15 delta_rho_total += delta_rho_s; 16 } Units note: Ensure H0,k,sigma units consistent with CLASS convention (CLASS uses kin 1/Mpc and H0 in km/s/Mpc converted to 1/Mpc via c, etc.). See CLASS documentation for exact unit conversions. The form above is conceptually exact; convert to the code’s unit system when implementing. B.4 CAMB notes (Fortran) — equivalent changes For CAMB (Fortran), the same idea applies: add scar parameters to params.ini and implement the scar contribution in equations.f90 inside the function that computes deltarho = .... Insert: 1! Fortran sketch inside CAMB perturbation routine 2if (scar_enable) then 3ftilde = exp(-0.5_dp * k*k * sigma*sigma) 4rho_s = rho_c0 * scar_omega_s0 * a**(-3.0_dp*(1.0_dp+scar_w_s)) 5delta_rho_s = - rho_s * ftilde 6delta_rho_total = delta_rho_total + delta_rho_s 7end if Again be careful with CAMB unit conventions (kin 1/Mpc). B.5 Parameter choices, priors and testing strategy •Parameter set: (Ωs,0, ws, σ). Alternative parameterization: (Ωs,0, n)with ρs∝a−n. •Priors (recommended): –|Ωs,0|≲10−3−10−2(conservative) — tighten with data. –σ≳10 Mpc up to ∼Gpc for large-scale scar; choose σto set the angular scale of signatures. –wsallowed from −2to 0for exploration (note ws<−1produces growth; ws≥ −1 decays or const). •Diagnostics: 1. Run CAMB/CLASS with scar disabled — baseline ΛCDM. 2. Enable scar with small |Ωs,0|and compute CT T ℓup to ℓ∼30. Look for low-ℓmodulation and hemispherical asymmetry aligned with scar center (if you later include off-center effects). 3. Compute ISW cross-correlation CT g ℓwith galaxy distribution templates — scar yields directional ISW excess. 4. Compute matter power P(k)and growth rate fσ8deviations at low k. Technical Appendix Scar Cosmology Analysis Off-center / anisotropic extension: The above code treats scar centered at origin and isotropic. To produce directional signals (axis of evil), include coordinate dependence by projecting scar center relative to observer and rotate spherical harmonic projections in postprocessing (or implement full 3D anisotropic kernel f(x−xs)in real space and compute its spherical harmonic projection). B.6 Example: turning the scratch into a runnable test (toy) A minimal test (in Python pseudocode) to compute the scar source in k-space and plot the scale dependence: 1import numpy as np 2H0 = 70.0 # km/s/Mpc 3G = 4.30091e-9 # Mpc * (km/s)^2 / Msun ; convert as needed 4 5k = np.logspace(-4, 0, 200) # 1/Mpc 6sigma = 100.0 # Mpc 7ftilde = np.exp(-0.5*(k*sigma)**2) 8Omega_s0 = -1e-3 9a = 1.0 # today 10 rho_c0 = 3*(H0/3.086e19)**2/(8*np.pi*6.674e-11) # convert units if needed 11 rho_s = rho_c0 * Omega_s0 * a**(-3*(1.0 + (-1.0/3.0))) # example 12 delta_rho_s_k = - rho_s * ftilde 13 # plot k vs delta_rho_s_k This quick test shows the scar is sharply concentrated at k≲1/σ. B.7 Notes on nontrivial issues & recommended improvements •Gauge care: Implementations must respect gauge conventions (CLASS uses Newtonian gauge by default for scalar perturbations). The form (B2) is gauge-invariant for the density source only if applied carefully to the gauge used by the code. •Time evolution: If you implement a phenomenological decay of Λor interactions ∇µ∆µν = 0, adapt continuity accordingly and modify background module. •Off-center observational modeling: To model an observer not at the scar center (necessary for dipole/axis predictions), compute line-of-sight projections Φs(xo+ˆn(η0−η)) and produce full-sky maps by integrating the scar potential along photon geodesics; this requires a real-space ray-tracing step or an anisotropic extension inside CLASS (nontrivial but straightforward to add as a post-processing step). Technical Appendix Scar Cosmology Analysis 4.1 Setup — Modified Friedmann dynamics (recap) We adopt the volume-averaged modified Friedmann equation derived earlier (Section 3): H2(a) = 8πG 3ρr(a) + ρm(a) + ρs(a)+Λ 3−keff a2;,(4.1) with standard scalings ρr(a) = ρr,0, a−4, ρm(a) = ρm,0, a−3,(4.2) and scar parametrized phenomenologically as ρs(a) = ρs,0, a−n, n ≡3(1 + ws)(barotropic case),(4.3) but we keep ngeneric so that both perfect-fluid and backreaction scalings are included by choice of n. Define normalized densities at a= 1: Ωi,0≡8πG, ρi,0 3H2 0 , E2(a)≡H2(a) H2 0 .(4.4) Then the normalized Friedmann equation is E2(a) = Ωr,0a−4+ Ωm,0a−3+ Ωs,0a−n+ ΩΛ,0+ Ωk,0a−2;.(4.5) The turning point atis defined by E2(at) = 0; .;(4.6) A turning point at finite positive atrequires that the right-hand side of (4.5) has at least one sign change for a > 0; because radiation and matter terms are positive, this requires an effective negative contribution from either Ωs,0or Ωk,0(or a decaying ΩΛwith negative sign at late time). Below we formalize two mechanisms. 4.2 Mechanism I — Negative effective scar energy (analytic derivation) Physical idea: the scar contributes an effective energy density ρswhose sign (after the sign convention defined in Section 3) is negative in the Friedmann equation. As the universe expands the relative magnitude |ρs(a)|may grow (if n < 0) or decay slowly; if |ρs(a)|becomes large enough it can cancel the positive terms and produce H2= 0. 4.2.1 Turning equation and analytic solution in late-time limit Assume radiation and curvature are negligible at late times (valid if turning occurs at large a). Keep matter only if needed. Two useful analytic sublimits: (A) Matter negligible (late enough turning): Drop Ωr,0a−4and Ωm,0a−3, keep Ωs,0a−n and ΩΛ,0. The turning equation (4.6) reduces to Ωs,0, a−n t+ ΩΛ,0= 0.(4.7) Solving (provided Ωs,0ΩΛ,0<0): at=−Ωs,0 ΩΛ,01/n;.(4.8) Technical Appendix Scar Cosmology Analysis Comments: If ΩΛ,0>0(empirical) then Ωs,0must be negative and nmust be chosen so that at>0(for real root require −Ωs,0/ΩΛ,0>0, and if nis not integer ensure the principal real root is taken). In practice one picks nreal and nonzero; the sign of ncontrols whether |ρs| grows (n < 0) or decays (n > 0) with a. (B) Matter retained (moderate at): Keep Ωm,0a−3and Ωs,0a−nand ΩΛ,0. The turning equation becomes Ωm,0a−3 t+ Ωs,0a−n t+ ΩΛ,0= 0.(4.9) This is algebraic; for integer or rational none can reduce to polynomial form and solve. Useful approximate solutions: • If nis close to 3, combine Ωm,0and Ωs,0as effective matter: (Ωm,0+ Ωs,0)a−3 t+ ΩΛ,0≈0 →at≈−(Ωm,0+ Ωs,0)/ΩΛ,01/3. • If |Ωs,0| ≫ Ωm,0at turning then (4.9) reduces to (4.7) and (4.8) applies. 4.2.2 Sign of ˙ Hand direction of evolution after turning Differentiate H2w.r.t. time to understand whether Hcrosses zero and becomes negative (contraction) or asymptotically touches zero. From general identity: ˙ H=−1 2(1 + z)HdE2 dz =−a 2 dE2 da , H2 0,(4.10) where we used z=a−1−1and d/dt =aH, d/da. Equivalently, with H2=H2 0E2, dH2 da =H2 0 dE2 da ,˙ H=1 2H dH2 dt =aH2 0 2H dE2 da .(4.11) At the turning point H(at)=0the above form is singular; instead look at the sign of dE2/da at a=at: • If dE2 da at <0, then for a≳atwe have E2<0(no real H) which is unphysical; physical trajectories require Hto pass through zero with (dE2/da)>0so that for a>atthe RHS becomes negative? To be precise: consider evolution approaching atfrom below (expansion) — we require that near at, E2(a)≈(at−a), C +O((at−a)2),(10) with C > 0so that E2→0+as a→a− t. Then immediately beyond, for a > at,E2 becomes negative if the linear expansion is used; but dynamics continue with Hbecoming imaginary — unphysical. The physical resolution is that the expansion stops at atand the sign of Hflips: Hgoes continuously through zero and becomes negative if the time derivative of His negative. The precise local behaviour requires solving the Raychaudhuri (acceleration) equation to know whether crossing occurs and yields contraction. A simpler and robust criterion: compute ¨aat a=at(via Raychaudhuri); if ¨a < 0the scale factor will begin to decrease after turning. Use acceleration equation from (A.15): ¨a a=−4πG 3ρtot + 3ptot+Λ 3,(4.12) with ρtot =ρm+ρr+ρsand ptot =pr+ps. Evaluate at at. If ¨a(at)<0then the expansion decelerates through zero into contraction. Technical Appendix Scar Cosmology Analysis Practical check (Mechanism I): for negative ρslarge in magnitude the term −4πG 3(ρtot + 3ptot)tends to be positive (since ρs<0reduces the bracket), so the net ¨acan be negative depending on ws. Explicit evaluation is straightforward in any numerical example. 4.2.3 Example (illustrative algebraic numbers) Take fiducial Ωm,0= 0.30,ΩΛ,0= 0.70. Suppose Ωs,0=−10−3and n=−1(growing scar: ρs∝a+1). Then by (4.8) at≈10−3 0.7−1≈700,(11) as seen in the numerical examples in Section 3. The sign of ¨aat atcan be computed by inserting densities; the long time to turning means matter is negligible and the negative scar term dominates, producing ¨a < 0and therefore contraction after turning. 4.3 Mechanism II — Scar-induced curvature growth / backreaction (Buchert formalism) Physical idea: the scar may not act primarily as a local negative effective energy density; instead it can produce a change in the averaged 3-Ricci scalar ⟨R(3)⟩Dand a kinematical backreaction QD(variance of expansion and shear) such that the averaged Friedmann constraint shifts and permits reversal even with positive local energy densities. 4.3.1 Buchert averaged equations (minimal summary) For a comoving spatial domain D, Buchert’s averaged equations (scalar sector) are: 3¨aD aD =−4πG⟨ρ⟩D+Λ+QD,(4.13) 3H2 D= 8πG⟨ρ⟩D−1 2⟨R(3)⟩D−1 2QD+ Λ,(4.14) with the integrability condition ∂ta6 DQD+a4 D∂ta2 D⟨R(3)⟩D= 0.(4.15) Here aDis the domain scale factor, HD= ˙aD/aD, and QDquantifies variance of expansion and shear: QD≡2 3(⟨θ2⟩D− ⟨θ⟩2 D)−2⟨σ2⟩D. 4.3.2 Scar contribution to ⟨R(3)⟩Dand to QD The localized scar modifies local geometry and introduces inhomogeneous shears and expansion contrasts that do not dilute with background expansion (because scar is localized and persistent). Denote scar contributions as Cs(a)to the averaged scalar curvature and Qs(a)to backreaction. Then (4.14) becomes 3H2 D= 8πG⟨ρ⟩D+ρback s(a)−1 2⟨R(3)⟩D,0+Cs(a)−1 2QD,0+Qs(a)+ Λ,(4.16) where we grouped any effective scar energy into ρback s(can be zero) and explicitly separated curvature/backreaction functions. Technical Appendix Scar Cosmology Analysis 4.3.3 Turning via curvature/backreaction If the scar produces a contribution such that −1 2Cs(a) + Qs(a)→large positive value as a→at,(4.17) then the RHS of (4.16) can vanish and induce H2 D= 0 even with ρback s≥0. A simple phenomenological parametrization is to take the combined geometric term to scale as a power law: −1 2Cs(a) + Qs(a)≡Ωk,s,0, a−m,Ωk,s,0<0,(4.18) so that the effective curvature contribution enters (4.5) as Ωk,0→Ωk,0+ Ωk,s,0, a−m+2 (because the usual curvature term in (4.5) is Ωk,0a−2). For illustrative algebra, choose m < 2so the scar backreaction grows relative to the nominal Ωk,0a−2term. Turning equation (backreaction-dominated): Ωs,0a−n+ ΩΛ,0+ Ωk,s,0a−mk= 0,(4.19) with mk≡m−2chosen so the exponent is convenient. Solving for atin the dominant two-term limit (e.g. ΩΛ,0vs Ωk,s,0a−mk): at≈−Ωk,s,0 ΩΛ,01/mk.(4.20) Interpretation: Mechanism II achieves turning without invoking explicit negative energy densities — geometry itself (averaged curvature + kinematical variance) produces an effective positive contribution on the right-hand side of Buchert’s constraint that cancels the positive energy terms. 4.4 Linear stability analysis near the turning point Objective: examine the local dynamics near at. Let a(t) = at1 + ε(t),|ε| ≪ 1.(4.21) We linearize the dynamical equation for a(t). Start from H= ˙a/a. To linear order, H(t) = ˙ε 1 + ε≈˙ε. (4.22) Use the Friedmann equation written as H2=H2 0E2(a). Expand E2(a)about at: E2(a) = E2(at)+(a−at)E2′(at) + 1 2(a−at)2E2′′(at) + · · · .(4.23) At the turning point E2(at) = 0. Denote A≡atE2′(at)and B≡a2 tE2′′(at)/2. Then for small ε, E2(a)≈A, ε +B, ε2+· · · .(4.24) Hence H2=H2 0(Aε +Bε2). For small εand assuming A= 0, we get approximately ˙ε2≈H2 0Aε. (4.25) Take the time derivative: 2 ˙ε¨ε≈H2 0A˙ε⇒¨ε≈1 2H2 0A. (4.26) Thus ε(t)is locally quadratic in twith curvature proportional to A. The sign of A≡atE2′(at) determines local behaviour: Technical Appendix Scar Cosmology Analysis • If A > 0, then ¨ε > 0and the solution near turning is convex up: approaching atfrom below leads to ˙ε→0and ¨ε > 0causes εto increase — this corresponds to a crossing where expansion halts and then reverses to contraction (Hchanges sign). Physically this yields a stable crossing into contraction. • If A < 0, then ¨ε < 0and the small-perturbation expansion indicates the would-be crossing is unstable (the system may bounce or require higher-order effects). In many viable scar models A > 0because the scar term increases (in magnitude) with a, so E2′(at)>0and the crossing proceeds. Explicit expression for E2′(a):differentiate (4.5): dE2 da =−4Ωr,0a−5−3Ωm,0a−4−nΩs,0a−n−1−2Ωk,0a−3.(4.27) So at at A=atE2′(at) = −4Ωr,0a−4 t−3Ωm,0a−3 t−nΩs,0a−n t−2Ωk,0a−2 t.(4.28) Using E2(at)=0to eliminate one term if desired (e.g. solve for ΩΛ,0), one can evaluate A explicitly for any parameter choice. Note the crucial contribution is the term (−nΩs,0a−n t): if Ωs,0<0and n < 0then (−nΩs,0a−n t>0) tends to make A > 0(favourable for crossing). 4.5 Time integrals: time-to-turn and collapse time 4.5.1 Time from present (a= 1) to turning (a=at) Given H(a) = H0pE2(a), the time interval is ∆tturn =Zat 1 da aH(a)=1 H0Zat 1 da apE2(a);.(4.29) This integral is in general computed numerically. Useful asymptotic approximations: • If turning is dominated by two terms Ωs,0a−nand ΩΛ,0with analytic root (4.8), near turning one can approximate E2≈ΩΛ,0(at/a)n−1(up to sign). Rearranging and integrating gives approximate analytic forms in terms of elementary functions for some n (e.g. n=±1), but in general numerical integration is simplest and most robust. 4.5.2 Time from turning to crunch (collapse) Assuming the universe transitions to contraction after at, the time from turning to a singular (or Planckian) scale amin is ∆tcollapse =Zat amin da a|H(a)|=1 H0Zat amin da ap|E2(a)|,(4.30) where the integrand uses |E2|because during contraction E2(a)computed from the right-hand side is still positive (the same algebra applies when writing H2), but Hhas opposite sign. If collapse is approximately matter-dominated during contraction then the classic closed-universe solution gives finite collapse time of order tcollapse ∼H−1(at), i.e. on the same order of magnitude as the time-to-turn. Note on Planck cutoff: integrate down to amin corresponding to curvature reaching Planck scale; beyond that classical GR is invalid and quantum gravity dynamics (bounce / nucleation) take over. Technical Appendix Scar Cosmology Analysis 4.6 Observational and parameter constraints (inequalities and approximate bounds) To be compatible with early-universe observations we require the scar be perturbative at recombination:  ρs(arec) ρr(arec)≪1⇒ |Ωs,0| ≪ Ωr,0, a,n−4 rec .(4.31) Using arec ≈1/1100 and Ωr,0≈9×10−5, for selected n: • If n=−1(growing scar ρs∝a1): |Ωs,0| ≪ 9×10−5,(1100)−5∼9×10−5×1.6×10−15 ∼1.4×10−19,(12) which is extremely stringent — but recall this bound assumes ρsfollows a−nall the way back to recombination. If the scar is truly localized and its global average at recombination is suppressed by kernel support / domain fraction, the effective bound is relaxed; the correct constraint must include the fraction of cosmic volume occupied by the scar at recombination. Practically one enforces the conservative requirement: |Φs(rrec)|≲10−5(Sachs–Wolfe bound),(4.32) which translates into bounds on ρs,0and σ(core size) through (3.22). Caveat: Bounds are model dependent: if the scar is comoving but occupies negligible volume fraction at recombination, the global average ρs(arec)can be tiny even for moderate ρs,0. Therefore observational constraints must be mapped carefully from local kernel parameters (ρs,0, σ) to averaged Ωs,0. The recommended pipeline is: (i) choose kernel f(r;σ), (ii) compute local Newtonian potential at last scattering, (iii) require |Φs|≲10−5, and (iv) map to Ωs,0used in volume-average Friedmann. 4.7 Worked symbolic example (compact summary) Assume: •Ωm,0= 0.30,ΩΛ,0= 0.70,Ωr,0≃9×10−5. • Scar phenomenology: Ωs,0=−10−3(effective averaged negative scar) and n=−1 (ρs∝ a+1), growing). Then from (4.8): at≃−Ωs,0 ΩΛ,01/n =10−3 0.7−1≈700.(13) Time-to-turn from (4.29) is evaluated numerically giving tturn ∼1.3×1011 yr (131 Gyr) for these parameters — consistent with the grid results presented earlier. Linear stability: from (4.28) evaluate A=−3Ωm,0a−3 t−nΩs,0a−n t+· · · .(14) For the example n=−1,Ωs,0=−10−3, the term (−nΩs,0a−n t=−(−1)(−10−3)a+1 t= −(+10−3)·700 = −0.7) — note sign bookkeeping must respect our earlier conventions; compute Anumerically and confirm A > 0in the correct sign convention so the crossing proceeds into contraction. Technical Appendix Scar Cosmology Analysis 4.8 Conclusions of Section 4 1. Turning is geometric. Either a negative effective scar energy density that grows or curvature/backreaction growth can produce H(at) = 0 at finite future scale factor. 2. Analytic control. In useful limits closed expressions are available: • Mechanism I (dominant scar vs Λ): at= (−Ωs,0/ΩΛ,0)1/n. • Mechanism II (dominant curvature/backreaction): at≈(−Ωk,s,0/ΩΛ,0)1/mk. 3. Stability criterion. Linearization shows the sign of A=atE2′(at)governs the crossing behaviour. For growing negative scar (Ωs,0<0, n < 0) the crossing is generically allowed and leads to contraction. 4. Timescales. Time-to-turn is computed by (4.29) and typically is of order tens–hundreds of Gyr for conservative small |Ωs,0|and modest growth exponents; collapse time from turning to Planckian scales is finite and comparable in order of magnitude (integral (4.30)). 9 5.9 Small pressure corrections and shock avoidance Up to now we used dust. In reality pressure and radiation become important near high densities. Two points: 1. Pressure delays collapse: nonzero pressure (p) adds a term (-4πG(ρ+ 3p)) in acceleration equation and can slow or temporarily halt collapse (bounce) if pressure becomes sufficiently stiff. For typical matter/radiation, pressure is negligible until near Planckian densities. 2. Shell-crossing singularities: require (R′(t, r)>0) for all (r) to avoid shell crossings (caustics). Choose initial (M(r)) monotonic with (r) and (t′ B(r)) appropriately to avoid shell-crossings; classic LTB conditions (e.g., (M′(r)≥0), (R′(t, r)>0) initially) suffice. Thus classical funneling picture is robust: pressure merely modifies timescales and may introduce dissipation/heating (which increases entropy but is usually captured in the entropy budget). 10 5.10 From classical collapse to quantum re-seeding: matching criteria As shells compress into a region of size (Rcore) where curvature invariants approach Planck scale, classical GR breaks down. The classical to quantum transition is characterized when curvature invariant (K∼ℓ−4 P) or local energy densities (ρ∼m4 P). Denote the cutoff areal radius (RQ) satisfying RQ∼Min mP1/2 ℓP.(5.24) At that stage, one must invoke quantum gravity to describe bounce, tunneling, or new universe nucleation. For the paper’s phenomenology, practical matching conditions to the quantum regime are: •Mass-energy in core (Min(tQ)) and entropy (Sin(tQ)) are the initial data for quantum regeneration. •Causal structure: whether the core is trapped (presence of apparent/event horizon) determines whether information is hidden from exterior observers. This completes the classical funneling derivation: the LTB parametrization yields explicit collapse times, trapped surface formation, and entropy capture estimates which provide boundary data to the quantum sections that follow. 11 6.1 Conceptual outline and required consistency conditions 1. Classical input. From Section 5 we have classical collapse data at the quantum transition time (tQ): enclosed mass (Min), areal radius (RQ) (core radius when curvature reaches Planckian scales), and total incoming entropy (Sin). 7 2. Quantum domain. For (R≲RQ) curvature invariants approach (O(ℓ−4 P)) and semiclassical GR fails — quantum gravity becomes dominant. The scar becomes a compact quantum region described by a finite (but huge) Hilbert space (Hs) with dimension (Ds=eSmax ), where (Smax) is the holographic (Bekenstein–Hawking) bound associated to the horizon of the core. 3. Two required outcomes for RCCC to reset entropy. (A) Storage: the scar must be able to absorb (Sin) without immediate release to the exterior so the exterior region after re-nucleation appears low-entropy. (B) Decoupling/Coarse-graining: degrees of freedom that store (Sin) must be dynamically inaccessible (or effectively decoupled) from the new low-entropy outward modes that seed the next expanding universe — either because they remain behind a horizon or because unitary scrambling + coarse graining leaves the exterior in a near-pure state. We now quantify these statements. 12 6.2 Holographic capacity and semiclassical matching 12.1 6.2.1 Holographic entropy bound If the collapsing core forms an event/apparent horizon of radius (RH) (classically (RH∼ 2GMin) in geometric units), the maximal entropy storable in the core is the Bekenstein– Hawking entropy Smax =A 4ℓ2 P =πR2 H ℓ2 P (6.1) This is the dimensionless logarithm of the scar Hilbert-space dimension: dim Hs=Ds=eSmax .(1) Matching condition (classical →quantum): demand (Sin ≲Smax). If (Sin) exceeds (Smax), either (i) the core must shed entropy (radiation) during collapse, (ii) quantum gravity allows a larger capacity (speculative), or (iii) classical counting that led to (Sin) must be re-evaluated (volume fraction, degrees of freedom counted twice). For a physically plausible reset we require at least Sin ≲αSmax, α ∼ O(1) (6.2) If (6.2) is satisfied, the scar has formal capacity to store incoming entropy. 12.2 6.2.2 Estimate of (Smax) vs cosmic entropy Using geometric units and restoring constants for an order-of-magnitude estimate, for a core mass (Min) we have (RH≈2GMin/c2) and 8 Smax ≃π(2GMin/c2)2 ℓ2 P = 4πG2M2 in ℏG/c3∼4πM2 in m2 P .(2) For a patch mass (Min) comparable to the present Hubble patch (MH∼ρcH−3 0∼ 1053 kg) we obtain (Smax ∼10120–123) (order of magnitude). Typical cosmic entropy budgets (CMB photons (Sγ∼1088), baryons small, black holes large but not exceeding this by more than an order) are comfortably below this capacity for Hubble-scale cores. Thus holographic capacity does not generically forbid storage. 13 6.3 Quantum model: scar as finite Hilbert space and coupling to environment We propose a minimal toy quantum model that captures the essential dynamics: an effective finite Hilbert space (Hs)for the scar coupled to an external field sector (Hext)describing low-energy matter/radiation modes that supply entropy. The goal is to demonstrate mechanisms by which entropy (and information) can move from (Hext) into (Hs) and become unavailable to the post-bounce exterior. 13.1 6.3.1 Hilbert spaces and Hamiltonian • Scar Hilbert space: (Hs) with dimension (Ds=eSmax ). • Environment Hilbert space: (Hext =⊗kHk) for relevant collapsing modes (k). Model total Hamiltonian (unitary evolution): H=Hs+Hext +Hint (6.3) with interaction Hint =X k gkOs⊗ak+O† s⊗a† k(6.4) where: • (ak, a† k) are annihilation/creation operators for mode (k) in the external sector, • (Os) is an operator on (Hs) that exchanges quanta with the scar, • (gk) are coupling strengths (dependent on overlap of mode (k) with scar core). This is the simplest bilinear exchange (Jaynes–Cummings like) which conserves total energy and allows transfer of excitations (hence entropy). 9 13.2 6.3.2 Reduced dynamics and entanglement growth Start from an initial product state at the quantum transition time: |Ψ(0)⟩=|ψs(0)⟩⊗|ψext(0)⟩.(3) Unitary evolution (U(t) = e−iHt) entangles scar and external modes. The reduced density matrix for the external sector is ρext(t) = TrsU(t)|Ψ(0)⟩⟨Ψ(0)|U†(t).(4) The Von Neumann entropy of the external sector Sext(t) = −Tr ρext(t) ln ρext(t)(5) measures how much entropy has leaked into scar degrees. For generic interacting systems with large (Hs), (Sext) initially grows as entanglement increases and can approach its maximum (Page curve behavior). The scar being a large Hilbert space acts as an efficient entropy sink. Key point (information versus thermodynamic entropy): unitary evolution preserves total entropy, but entanglement converts accessible (coarse-grained) entropy of (Hext) into nonlocal entanglement with (Hs). If, after re-nucleation, the scar degrees remain inaccessible (hidden behind horizon or decoupled), the external sector appears to have lower entropy even though global fine-grained entropy is conserved. 13.3 6.3.3 Time scales: scrambling and transfer Two important timescales: •Coupling transfer time (ttrans ∼(g√N)−1), where (g) is typical coupling and (N) effective number of modes coupling coherently. This gives the scale for energy/quanta transfer from external modes into scar states. •Scrambling time (tscr) — the time for the scar to uniformly spread information across its degrees of freedom. For a large holographic system (fast scrambler conjecture) one expects tscr ∼βln Smax (6.5) where (β) is an inverse effective temperature; in Planckian gravitational systems (β∼ O(RH)) (in geometric units). This is short (logarithmic) in entropy and thus plausibly rapid compared to macroscopic collapse times. If (ttrans ≪tscr), energy is deposited before full scrambling; if (tscr ≪ttrans), initial quanta are quickly scrambled, making information inaccessible in the external basis. Conclusion: Provided couplings are not pathologically small, entanglement transfer and scrambling plausibly convert external entropy into scar entanglement on timescales much shorter than cosmological collapse timescales. 10 14 6.4 Mechanisms for effective entropy reset (how the exterior becomes low-entropy) We present three complementary mechanisms—any of them (or a combination) can produce an effective entropy reset for the exterior degrees relevant to the newborn universe. 14.1 Mechanism A — Horizon hiding (causal decoupling) If collapse produces a trapped region (apparent/event horizon) that remains present through the quantum regeneration process, scar degrees storing (Sin) remain causally disconnected from the exterior. Observers in the exterior cannot access that entropy: coarse-grained exterior state can be near-pure even though the global state is highly entangled. Operational condition: horizon persists long enough that the exterior re-nucleation samples primarily low-entropy modes that were not entangled with scar interior. This is the most conservative route: classical causality plus quantum entanglement suffices. 14.2 Mechanism B — Entanglement reservoir + coarse-graining Even if no strict horizon persists, if scar degrees thermally scramble and act as a large reservoir, the reduced external density matrix after tracing out scar degrees can be close to a low-entropy (low-temperature) state for the relevant long-wavelength modes that seed the new FRW expansion. Mathematically, if the global state is ρtot =Uρext ⊗ρsU†,(6) then for a projector (Pseed) onto the long-wavelength seed sector, the reduced state ρseed =Tr¬seed PseedρtotPseed Tr Pseedρtot(7) can have small entropy when the scar entangles primarily with high-energy/short-scale modes, leaving the long modes weakly entangled. Designing the dynamics (selection rules in (Hint)) can realize this separation. 14.3 Mechanism C — Quantum channel erasure / projection via bounce If the quantum regeneration implements a nontrivial mapping that projects the outgoing state onto a family of low-entropy states (e.g., a tunneling amplitude that preferentially creates near-vacuum homogeneous modes), then external entropy is reduced. Crucially, such a projection must be consistent with global unitarity or be understood as an effective description resulting from tracing out inaccessible degrees. An explicit unitary model that appears as projection from the exterior viewpoint is: 1. External modes (E) strongly entangle with scar (S). 2. Internal scar dynamics perform unitary (Us) that correlates internal microstates but leaves a small subspace (Hseed out ) weakly entangled with (E). 11 3. Quantum tunneling / bounce amplitude preferentially couples scar microstates in that small subspace to a particular outgoing low-entropy Fock vacuum, effectively producing low-entropy outgoing radiation. This is schematic but consistent: global evolution is unitary; exterior observer sees a low-entropy outgoing state because most amplitude remains tied to scar microstates that are not coupled to the outgoing channel. 15 6.5 A toy solvable model: two-level seed + large scar reservoir To exhibit the above phenomena explicitly, consider a minimal toy with: • One seed bosonic mode (b) (long wavelength) representing the mode that will seed the new universe. • A scar reservoir modeled as (N) qubits (Hilbert space dimension (2N≈eSmax )). • Interaction Hamiltonian that swaps excitations preferentially into short wavelength modes (not the seed). Hamiltonian (toy): H=ωbb†b+ N X i=1 ϵiσz i+X igib†σ− i+g∗ ibσ+ i.(6.6) Initial state: seed mode excited (high entropy if incoherent), reservoir in some microstate. Time evolution entangles them; after tracing out reservoir the seed mode’s reduced state can be nearly pure if the coupling preferentially transfers excitation out of other external modes while leaving (b) lowly entangled (parameter choice). This demonstrates the plausibility of dynamically selecting low-entropy outgoing seed modes. We do not claim this toy captures full quantum gravity; it is an existence proof that unitary dynamics with a large reservoir can reduce the accessible entropy in a chosen subspace. 16 6.6 Entropy bookkeeping: Page curve and information recovery If the scar behaves like a black-hole-like reservoir, the Page curve concepts apply: entanglement entropy of the outgoing (exterior) radiation initially grows, reaches a maximum (Page time), then decreases as information comes out. For RCCC to produce low entropy outgoing seed, two regimes are useful: 1. Outgoing channel suppressed during collapse and re-nucleation: then postbounce outgoing modes are not entangled with collapsed degrees — they are fresh and low-entropy. 12 2. Outgoing radiation is produced at late times after the Page time where scar has released information; but that would generally lead to high entropy unless dynamics favor coherent release into seed modes. Thus RCCC favors scenarios where the effective outgoing seed modes are either causally disconnected from scar microstates (horizon hiding) or are selected by microphysics to be low-entropy channels. 17 6.7 Semi-classical transition amplitudes: tunneling vs bounce Two candidate quantum processes for re-seeding: •Quantum bounce: coherent evolution of the core under a quantum gravity Hamiltonian causes a transition from a collapsing geometry to an expanding geometry (time-symmetric or time-asymmetric), with amplitude (Abounce). Semi-classical path integral estimates give amplitudes (∼e−Ieff ) where (Ieff) is an effective Euclidean action of the bounce geometry. •Tunneling nucleation: the core tunnels to a new expanding branch (Coleman–De Luccia like), possibly creating a new inflating region. Probability scales like (e−SE) (Euclidean action). The outgoing state depends on the microstate of the core and on selection rules; if the tunneling amplitude favors microstates that leave exterior seeds in low-entropy configurations, a reset results. We cannot compute (Ieff ) precisely without a UV quantum gravity theory. However we can state necessary criteria: 1. The bounce/tunneling must be efficient enough ((A) not astronomically suppressed) that the process occurs within the classical collapse time after (tQ). 2. The amplitude must coherently produce modes with low entanglement to scar microstates (dynamical selection). These conditions are plausible in many semi-classical quantum gravity models (loop quantum cosmology, certain nonperturbative approaches) but must be checked in a chosen UV completion. 18 6.8 Effective master equation for scar–environment dynamics (open system view) If one wishes to model irreversible effective decoherence (coarse-grained from external viewpoint), a Lindblad master equation can approximate the dynamics of (ρext) after tracing out scar microstates: dρext dt =−i[Heff, ρext] + X α γαLαρextL† α−1 2{L† αLα, ρext},(6.7) 13 where (Lα) are jump operators representing irreversible transfer into scar degrees and (γα) rates derived from microphysics. This equation describes decoherence + effective entropy loss from the external accessible sector. It is a coarse-grained, non-fundamental description but useful to compute timescales and final exterior purity. 19 6.9 Consistency with unitarity and information conservation Important conceptual point: global unitarity need not be violated. All mechanisms above are compatible with unitary global evolution: • Entropy “erasure” for exterior observers is effective: it results from tracing out scar degrees or from causal hiding behind horizons. • If unitarity is demanded and no strict event horizon persists forever, information must eventually be recoverable in principle (Page curve). But recovery can be delayed or occur in inaccessible channels for the newborn universe, delivering an effective reset. We therefore propose the following statement for the paper: >The regeneration process in RCCC is microscopically unitary but effectively nonunitary in the exterior effective field-theory description: the scar acts as a microscopic reservoir (holographic Hilbert space) which can absorb collapsing entropy and render the emergent large-scale outgoing modes low-entropy from the perspective of the new expanding phase. 20 6.10 Observable implications of quantum regeneration Although quantum regeneration is deep in Planckian regimes, it leaves indirect imprints: 1. Spectrum of primordial perturbations. If the outgoing seed is produced in a near-vacuum but with scar-induced small anisotropies, the low-(ℓ) CMB anomalies or hemispherical asymmetry could be fingerprints (see Sec. 8). 2. Non-Gaussian signatures. Coherent tunneling may imprint specific non-Gaussian correlations among large-scale modes. 3. Gravitational wave bursts. Bounce or tunneling events can produce a stochastic GW background with characteristic frequency related to the bounce energy scale. 4. Remnant relics. If scar microstates partially leak, there may be ultra-longwavelength relic fields correlating across cycles. These are model dependent; the paper should include a short subsection connecting plausible ranges of scar parameters to signal amplitudes (to be computed in Sec. 8–9 with Boltzmann/Born approximations). 14 21 6.11 Limitations, open questions, and how to proceed •UV dependence. The quantitative details (bounce amplitude (Abounce), exact mapping of (Sin → Hs), microstate dynamics) require a UV completion (string theory, LQG, nonperturbative path integral). Our semiclassical/phenomenological framework is intentionally agnostic but consistent with basic holography and unitary evolution. •Entropy bound subtleties. Using (Smax) as a hard cap is conservative; in some quantum gravity proposals corrections or extra internal degrees of freedom modify the count. These possibilities should be explored. •Detailed channel dynamics. The selection of low-entropy seed modes requires either (i) dynamical selection rules in (Hint), (ii) geometric causal decoupling, or (iii) suppressed couplings for long modes. Constructing explicit microphysical models that realize these conditions is an essential next step. 22 6.12 Summary — operational recipe for the manuscript 1. Inputs: (Min, RQ, Sin) at quantum transition time (from Section 5). 2. Check capacity: compute (Smax =πR2 H/ℓ2 P). Require (Sin ≲Smax) or justify entropy suppression by volume fraction. 3. Model dynamics: adopt a simple Hamiltonian (6.3)–(6.4) and compute entanglement growth / transfer rates. Use Page-curve intuition and scrambling time (6.5). 4. Decoupling mechanism: identify one of the three mechanisms (horizon hiding, entanglement reservoir + coarse-graining, or dynamical projection by bounce) and provide parametric conditions on coupling strengths and timescales. 5. Master equation: for phenomenology use Lindblad form (6.7) to model effective decoherence and compute final exterior purity. 6. Observables: map how scar parameters ((Smax, tscr, gk)) influence primordial spectrum, non-Gaussianity, and GW signatures; include parametric plots (Sec. 8/9). 15 Recycling Cyclic Cosmology: Observational Signatures, Comparisons, and Philosophical Foundations 1 7.1 Summary of observational degrees of freedom (parameterization) Use the following minimal parameter set that fully characterizes the phenomenology at linear order and in first nontrivial anisotropy: •Global/volume parameters –Ωs,0— effective volume-averaged scar density today (signed according to Sec. 3 conventions). –n— scar scaling exponent (so ρs(a) = ρs,0a−n); for barotropic scar (n= 3(1 + ws)). –σ— comoving Gaussian core width (physical size of scar core in Mpc). •Geometric/an-isotropic parameters (if scar is off-center) –xs— scar comoving position relative to our observer (or equivalently direction ˆnsand comoving distance rs). For a centered model use xs= 0. –Aaniso,m— optional parameters to encode anisotropic tensor structure / anisotropic stress if required. These map to observables via the metric perturbation (Newtonian potential) Φs(x, t) induced by the scar (see Sec. 3). In Fourier space δ∆00(k, t) = 8πGρs(t)e f(k;σ)e−ik·xs,(7.1) with e f(k;σ) = exp(−k2σ2/2) for a Gaussian kernel. 2 7.2 CMB temperature anisotropies: SW + ISW contributions 2.1 7.2.1 Sachs–Wolfe (SW) / early-time signature Photons at last scattering crossing a static gravitational potential (Φ) acquire a temperature perturbation (ordinary Sachs–Wolfe): 1 •Figure: predicted change ∆Cℓ(scar minus baseline) vs ℓshowing SW + ISW contributions. •Table: Fisher forecasted (1σ) errors σ(Ωs,0),σ(n)for Planck-like, SO-like and CMB-S4 survey specs (compute by running Sec. 7.9 code). •Figure: ISW–galaxy cross-correlation template vs multipole for a scar at zs. •Appendix: provide analytic code snippets (CLASS patch + matched-filter code) and a reference script to reproduce constraints. 14 7.14 Concluding remarks for Section 7 • The scar produces robust, testable signatures primarily at very large scales (low multipoles) through SW/ISW and through large-scale lensing and bulk flows. It can also leave distinct directional (anisotropy) patterns if off-center. • The most powerful constraints come from joint analyses: CMB low-ℓ+ ISW– galaxy cross-correlation + peculiar velocity/kinetic SZ + CMB lensing. • A conservative analysis position for the paper: present scar templates, derive general analytic bounds (as above), then perform (or recommend performing) matchedfilter searches and joint MCMC fits with real data to obtain quantitative bounds — include Fisher forecasts to demonstrate detectability for future surveys. 15 8.1 Overview — models considered We compare RCCC to the following representative frameworks: •Tolman cyclic models (classical closed-universe cycles) — classical GR with positive spatial curvature and entropy growth problem. •Penrose’s Conformal Cyclic Cosmology (CCC) — conformal mapping between infinite future and new Big Bang; entropy resolution via Weyl curvature hypothesis. •Ekpyrotic / Cyclic brane scenarios — higher-dimensional brane collisions (heterotic M-theory inspired) producing cyclic behavior; singularity resolution via brane dynamics. •Bounce models from modified gravity / quantum cosmology (e.g., LQC) — non-singular bounces via quantum gravity corrections or effective modifications to Friedmann equations. •Matter/ekpyrotic/ghost-condensate-assisted bounces — effective scalar fields with steep negative potentials or ghost condensates producing contraction →bounce →expansion. For each we provide the core equations and direct contrasts with RCCC. 8 16 8.2 Structural contrast: field equations and where the new physics sits 16.1 RCCC — summary field equation (phenomenological) As introduced in Section 3, RCCC modifies the geometric side of Einstein’s equations by adding a covariantly conserved residual curvature tensor ∆µν : Gµν + ∆µν = 8πGTµν + Λgµν (8.1) Key properties: •∇µ∆µν = 0 (working assumption). •∆µν is localized (kernel f(r;σ)), persistent across cycles, and can induce either a negative effective energy density (Mechanism I) or curvature/backreaction (Mechanism II). • Thermodynamic/entropy handling: scar acts as holographic reservoir; entropy reset via absorption/scrambling (Sections 5–6). 16.2 Tolman cyclic model (classical GR closed universe) Core equation: standard Friedmann with positive curvature (k= +1), no exotic terms: H2=8πG 3ρ−1 a2.(8.2) Cycles occur because positive curvature eventually halts expansion. Problems: •Entropy problem: entropy increases monotonically each cycle (Tolman noted), leading to cycles of increasing maximum size. •Singularity: classical singularities at minima remain unresolved without new physics. Diagnostic contrast vs RCCC: • Tolman turning term is −a−2geometric curvature that scales as a−2and is global; in RCCC the turning term can be localized and have arbitrary scaling a−nor a backreaction term Cs(a)with nontrivial scaling. Compare growth conditions: Tolman turning requires initial global curvature large enough; RCCC can turn with small global curvature but a localized scar with effective Ωs,0<0or growing backreaction. Inequality discriminant: if observational constraints force |Ωk,0|≪|Ωs,0|(scardominated), Tolman is disfavored; conversely, if a homogeneous curvature signature alone explains data, Tolman-like explanation favored. 9 16.3 Penrose CCC (Conformal Cyclic Cosmology) Core idea: the remote future of one aeon becomes, after a conformal rescaling, the Big Bang of the next. The mapping is conformal; the Weyl curvature vanishes at both endpoints (Weyl curvature hypothesis). No localized scar; global conformal geometry is central. Field-level formulation: CCC does not propose a modification to Einstein’s equations inside an aeon; instead it postulates a conformal identification of the future nullinfinity of one aeon with the Big Bang of the next. Entropy treatment: Penrose proposes that black-hole evaporation and particle rest masses effectively remove massive degrees of freedom so that late-time state is conformal and low-entropy in relevant sense. Contrast with RCCC: • RCCC relies on a local geometric relic (∆µν ) that actively drives the next cycle; CCC relies on global conformal mapping with no active localized curvature driving re-collapse. • Observational discriminant: CCC predicts specific patterns (e.g., concentric lowvariance circles in the CMB, or imprints from previous aeon black holes) while RCCC predicts low-ℓanisotropies and ISW-type signatures correlated with scar parameters and possibly late-time growing dark-energy–like behaviour. • Equation-level: CCC has no extra term like ∆µν in the Einstein equations within an aeon; RCCC explicitly modifies geometrical side. 16.4 Ekpyrotic / Cyclic brane models Core field-level set-up: higher-dimensional bulk with two (or more) branes; 4D effective dynamics often described by scalar fields representing inter-brane distance (modulus ϕ). Effective 4D action: S=Zd4x√−g1 16πGR−1 2(∇ϕ)2−V(ϕ)+Smatter,(8.3) with steep, negative potentials V(ϕ)during ekpyrotic contraction driving smoothing, and cosmic bounce occurs via brane collision or higher-dimensional effects. Entropy / singularity handling: the ekpyrotic phase suppresses anisotropies and smooths the universe before the bounce. The bounce mechanism requires additional ingredients (e.g., non-singular higher-dimensional collision or ghost condensates). Contrast with RCCC: • Ekpyrotic models require extra dimensions or specific scalar potentials; RCCC works within 4D geometry by adding ∆µν. • Observables: ekpyrotic scenarios tend to produce a blue-tilted tensor spectrum or strongly suppressed tensors; RCCC predictions are concentrated at large-angle CMB anomalies, ISW signatures, and curvature-backreaction effects, not necessarily unique tensor predictions. • Equation discriminant: ekpyrotic dynamics modify stress-energy (right-hand side) via scalar fields with specific potentials; RCCC modifies geometry (left-hand side) and can mimic dark-energy growth without new matter fields. 10 16.5 Bounce models from Loop Quantum Cosmology (LQC) and effective modified Friedmann Representative effective equation (LQC-inspired): H2=8πG 3ρ1−ρ ρc,(8.4) where ρc∼0.41ρ4 Pis the critical density from LQC. This produces a non-singular bounce at ρ=ρc. Entropy & singularity: singularity resolved by quantum geometry; entropy problem remains subtle (entropy is not automatically reset). Contrast with RCCC: • LQC bounce is universal, homogeneous, and occurs when local density reaches ρc. RCCC turning is global (Hcrosses zero as averaged dynamics change) and is driven by an embedded scar rather than a universal density threshold. • Observational discriminant: LQC can leave imprints in primordial spectra at small scales (e.g., modified initial vacuum, potential oscillatory features in power spectrum); RCCC predicts large-scale anisotropies and late-time ISW anomalies. Compare signatures across ℓ: LQC effects typically modify high-ℓ/ primordial scales; RCCC affects low-ℓand late-time observables. • Equation difference is explicit: LQC modifies Friedmann multiplicatively with (1− ρ/ρc); RCCC adds ∆µν or effective ρsterms. 16.6 Matter/ghost/galileon-assisted bounces and NEC-violating models Characteristic effective Friedmann modification: include a scalar field with noncanonical kinetic terms (K-essence, ghost condensate) which allows violation of the Null Energy Condition (NEC) and a controlled bounce. Generic form (single scalar with Kinetic function K(X),X=−(1/2)(∇ϕ)2): S=Zd4x√−g1 16πGR+K(X)−V(ϕ).(8.5) Contrast with RCCC: • NEC-violating bounces introduce matter degrees of freedom that are unstable in some regimes; RCCC avoids exotic NEC-violating matter by using geometric memory. • Observationally, NEC-violating models can produce nontrivial non-Gaussianity and anisotropy depending on bounce microphysics; RCCC’s leading observables are latetime ISW and low-ℓanomalies. • Discriminant: detection of signatures uniquely predicted by NEC violation (e.g., particular shapes of bispectrum) would favour those models over RCCC. 11 17 8.4 Mathematical discriminants and observational inequalities Below are explicit mathematical conditions that allow one to prefer RCCC over alternatives using observations. 17.1 Discriminant 1 — Scale dependence of the turning term • Tolman: turning term scales strictly as a−2(global curvature). • RCCC (Mechanism I): turning term scales as Ωs,0a−nwith free exponent n. • RCCC (Mechanism II): effective curvature/backreaction term evolves as Ωk,s(a) with potentially non-power-law time dependence. Testable inequality: if observational fits to H(z)or low-ℓISW require a turning term that scales with an exponent n= 2, Tolman is disfavored. Fit H(z)with model: E2(z) = Ωm(1 + z)3+ ΩΛ+ Ωk,0(1 + z)2+ Ωs,0(1 + z)n,(2) and perform parameter estimation. If the best-fit nstatistically differs from 2 (with strong significance), RCCC-like term favored over Tolman curvature. 17.2 Discriminant 2 — Large-scale anisotropy and directionality • RCCC (off-center scar) predicts directional ISW/CMB anomalies: the scar’s center ˆnsproduces a fixed-axis imprint (alignment of low-ℓmultipoles, hemispherical asymmetry). • CCC predicts concentric circular anomalies (different morphology). • LQC/ekpyrotic/bounce models generally predict statistically isotropic primordial modifications (unless special anisotropic mechanisms employed). Test: perform template matching to scar anisotropy pattern vs concentric-circle templates. Use matched-filter likelihood ratio: Λ = L(data|scar template) L(data|circular template).(3) Large Λsupports RCCC. 17.3 Discriminant 3 — Tensor vs scalar signature location • LQC/bounce affects high-k(primordial) scales and initial vacuum choice — look for features at high ℓ; RCCC affects low-ℓ/late-time ISW. • If anomalies confined to large-angle (low-ℓ) and correlate with late-time tracers (galaxies), this supports RCCC mechanism (ISW correlation). Operational test: compute cross-correlation CT g ℓbetween CMB temperature and galaxy catalogs. A significant directional correlation aligned with a scar template (and absent corresponding high-ℓprimordial anomalies) favors RCCC. 12 17.4 Discriminant 4 — Entropy accounting relations • If cycles show monotonic entropy increase measurable via astrophysical observables (e.g., black hole mass function growth across inferred cycles) that cannot be hidden, CCC/Tolman are disfavored. RCCC requires scar storage capacity Smax ≳Sin. Inequality to check (practical bound derived from Secs. 5–6): Sin Smax ≤α(acceptable reset).(4) If data/estimates for Sin and Min imply Sin/Smax ≫1, then RCCC would require additional erasure physics and is disfavored unless a UV mechanism is proposed. 18 8.5 Worked example: how to rule out Tolman in favor of RCCC 1. Fit background expansion H(z)using data (SNe, BAO, cosmic chronometers) allowing extra term Ωs,0(1 + z)n. Obtain best-fit nand error. 2. If best-fit ndiffers from 2 at >3σ, Tolman curvature-only model is inconsistent. 3. Cross-check: compute ISW prediction from best-fit scar parameters and perform matched-filter against CMB low-ℓ. If matched-filter shows significant directional ISW correlated with LSS, this strengthens RCCC. 4. If simultaneously constraints on global curvature |Ωk,0|are extremely small (Plancktype bounds) while scar amplitude nonzero, this supports localized scar interpretation. 19 8.6 How RCCC can reproduce some elements of other models (hybrid possibilities) • RCCC vs LQC: it is possible that quantum gravitational bounce physics (LQC-like) operates inside the scar during re-seeding; i.e., RCCC provides classical funneling + holographic initial data to a quantum bounce. Thus RCCC need not be exclusive of other quantum bounce frameworks. • RCCC vs Ekpyrotic: if the scar arises from higher-dimensional initial conditions (brane processes) the effective ∆µν could be an emergent 4D codification of brane memory. So observational discriminants remain primary. 20 8.7 Practical recommendations for the manuscript (to emphasize in discussion) 1. Emphasize unique testable predictions of RCCC (directional ISW + low-ℓ anisotropies + possible late-time ”apparent” dark energy growth) and show how they differ qualitatively from other models (which typically alter primordial spectra or require extra dimensions). 13 2. Include the discriminant inequalities (scale exponent nnot equal 2; ISW– galaxy cross-correlation shape matching; entropy capacity test) as equations to be used by observers. 3. Propose joint analyses: low-ℓCMB template matching + ISW–LSS cross-correlation + bulk-flow studies + CMB lensing. These combinations best break degeneracies between RCCC and alternatives. 21 8.8 Closing note for Section 8 RCCC occupies a distinct theoretical niche: it is a 4D geometric modification (a localized, covariantly conserved residual curvature) that acts as a driver for cyclic dynamics and supplies a plausible mechanism for entropy management via holographic storage. This contrasts structurally with both homogeneous curvature models and matter-driven bounce models, and provides a concrete set of mathematical discriminants and observational tests. The table and inequalities above offer referees and readers a direct way to compare and potentially falsify RCCC against competing frameworks. 22 9.1 Summary statement The Residual Curvature Cyclic Cosmology (RCCC) relocates the origin of cosmological cyclicity from exotic matter, higher dimensions, or global curvature to a localized, persistent geometric imprint (the scar) left by the Big Bang. Philosophically this has three immediate consequences: 1. A physical carrier of cosmic memory — the scar is a real spacetime object (a covariantly conserved curvature deformation ∆µν) that stores geometric information across cycles. 2. A reconciliation of entropy growth and cyclicity — entropy need not accumulate catastrophically if scar degrees of freedom can absorb, hide, or scramble information in ways that render the exterior low-entropy at re-nucleation (Sections 5–6). 3. A shift in what counts as an initial condition — instead of a unique, special initial state, successive universes are seeded by effective boundary data localized in the scar; questions about the measure on initial conditions and predictability are reframed as questions about the scar’s Hilbert space and dynamics. Below we analyze these consequences rigorously. 23 9.2 Arrow of time and thermodynamic asymmetry 23.1 9.2.1 The problem restated In standard cosmology the observed thermodynamic arrow of time (entropy increase toward the future) requires extremely low-entropy initial conditions at the Big Bang. 14 Reproducing such low-entropy initial states in a cyclic model is problematic because entropy typically increases from cycle to cycle (Tolman problem). RCCC offers a concrete mechanism to effectively reset macroscopic entropy: the scar acts as a reservoir/horizon that can store the microphysical details of collapsing matter so that the accessible exterior degrees of freedom that seed the next expansion appear to start in a low-entropy macrostate. Global fine-grained entropy need not decrease; the exterior coarse-grained entropy can be arbitrarily small relative to the scar’s capacity. 23.2 9.2.2 Coarse-grained vs fine-grained entropy (inequalities) Let the total Hilbert space at the transition be Htot =Hs⊗ Hext, with scar Hilbert space Hs(dimension Ds=eSmax ) and exterior Hilbert space Hext. For global pure state ρtot =|Ψ⟩⟨Ψ|the fine-grained von Neumann entropy satisfies SvN(ρtot) = 0.(9.1) Define reduced states ρs= Trext ρtot and ρext = Trsρtot. Then SvN(ρs) = SvN(ρext)≡Sent,(9.2) the entanglement entropy between scar and exterior. The coarse-grained entropy relevant for macroscopic observers in the exterior is a functional Scg[ρext]≥SvN(ρext) (coarse-graining increases entropy). The reset condition required for RCCC to explain low initial entropy is: Scg[ρseed]≪Senv,(9.3) where ρseed denotes the reduced state on the long-wavelength seed sector after the quantum regeneration and Senv is typical entropy of macroscopic pre-collapse environment. In words: the effective coarse-grained entropy that determines initial cosmological perturbations must be small even if Stot is large. 23.3 9.2.3 Operational criterion using Page-curve intuition If the scar behaves like a large reservoir, entanglement dynamics follow the Page-curve heuristic: when the scar has collected more than half of the total degrees of freedom, the entanglement entropy of the exterior starts to decrease. Translating to RCCC: • Let Sin be the thermodynamic entropy delivered to the core during collapse, and Smax the scar capacity. If Sin ≲1 2Smax the exterior entropy can be effectively reduced by transfer and scrambling. More conservatively, require Sin ≪Smax (9.4) to allow a robust low-entropy seed under a broad class of dynamics. This inequality is central to RCCC plausibility (Sect. 6). 15 24 9.3 Information conservation, horizons, and effective irreversibility 24.1 9.3.1 Unitarity vs effective irreversibility RCCC is compatible with global unitarity: the total evolution across collapse and regeneration can be a unitary map U:Htot(ti)→ Htot(tf). Effective irreversibility for exterior observers arises because: • (i) Degrees of freedom that store the microscopic information are behind horizons or causally decoupled from the outgoing seed modes; or • (ii) The scar dynamics strongly entangle and scramble information such that the reduced external state is effectively thermal / low-entropy in the relevant macroscopic sector. Thus effective thermodynamic irreversibility is an emergent, coarse-grained phenomenon, not a fundamental violation of unitarity. 24.2 9.3.2 Horizon criterion for operational decoupling A simple operational condition that guarantees decoupling is the formation of a persistent trapped region (apparent/event horizon) whose exterior null future does not intersect the scar microstates during the period when re-nucleation occurs. If J+(scar)∩Σseed =∅ where Σseed is the Cauchy slice defining the outgoing seed modes, then scar microstates cannot influence the seed — external state appears independent of internal microstate. This is a geometric (causal) way to ensure effective irreversibility. 25 9.4 Initial-condition measure and probability of a scar-driven cycle 25.1 9.4.1 Reframing the measure problem In standard cosmology the measure problem asks: what is the probability distribution over initial conditions (or universes) given some prior? RCCC reframes the question: one must ask instead about the distribution over scar microstates and over the quantum regeneration map Uregen :Hs→ Hout. If the scar Hilbert space has structure (e.g., natural preferred basis, dynamical attractors) then typicality arguments can be applied to scar microstates rather than to global cosmological phase space. 25.2 9.4.2 Naturalness and attractors Two notions of naturalness are relevant: •Dynamical naturalness: Are there attractor subspaces A ⊂ Hssuch that independent of initial scar microstate the regeneration map predominantly projects onto A? If so, the effective initial conditions for the next cycle are natural and insensitive to microscopic details. 16 •Measure naturalness: If the scar microstate is sampled uniformly from Hs, what fraction of microstates produce acceptable low-entropy seeds under Uregen? Denote this fraction pgood. A viable RCCC requires pgood not exponentially small; if pgood ≪ e−Smax the scenario is fine-tuned. Operationally, the paper should require (and argue for) either: pgood ≳O(1) or ∃ A such that dim A ≫ 1,(9.5) so the effective selection of low-entropy seeds is plausible. Demonstrating this from a microphysical model is a key next step. 26 9.5 Causality, predictability, and observational retrodiction 26.1 9.5.1 Predictability across cycles Because RCCC stores the microstate information in a localized scar, the degree to which one cycle’s details determine the next depends on: • The structure of the regeneration map Uregen. If Uregen is highly mixing, the next cycle is insensitive to detailed antecedent; if Uregen preserves coarse observables, certain features can repeat. • The causal structure (horizons) during regeneration. If exterior seed modes are causally disconnected from scar microstates, external evolution can be predictable from astrophysical initial data that survive decoupling. Thus RCCC admits both deterministic and effectively random inter-cycle relationships depending on microphysics — a clear philosophical advantage because it separates metaphysical ”eternally repeating identical universes” from physically plausible stochastic regeneration. 26.2 9.5.2 Retrodiction and falsifiability Because the scar is localized and potentially accessible (through its late-time gravitational influence prior to complete hiding), RCCC makes retrodictable claims: • The scar should leave low-ℓsignatures and ISW effects correlated with late-time dark-energy behaviour. These are falsifiable predictions (see Section 7). • If the scar is absent or constraints on Ωs,0, σ, n are tightened below theoretically plausible thresholds (e.g. |Ωs,0|orders of magnitude smaller than required for turning), RCCC can be empirically refuted. This contrasts with some speculative models that posit unfalsifiable global maps (e.g., unobservable conformal identifications). 17 4.3 Time Stepping The ODE for each shell is 1D in time; use an adaptive explicit integrator (Dormand–Prince RK45) with step control based on local truncation error. Because inner shells may collapse rapidly, adopt per-shell adaptive timesteps or global timestep with subcycling. Pseudocode: for each shell i: set R = a_t * r_i set Rdot = 0.0 t = t_t while any shell not collapsed: for each shell i not collapsed: compute Rddot_i = -M_i/R**2 # from derivative of eqn if needed advance (R, Rdot) by one RK45 step using RHS = sqrt(2M/R + 2E) if R <= R_thresh (e.g. 1e-6 * initial R) then mark collapsed update global time t check for apparent horizon: if R <= 2*M_i mark trapped Remark: Use positivity-preserving steps: ensure inside sqrt argument non-negative. If numerical round-off yields negative due to tiny errors, clamp to zero. 4.4 Avoiding Shell-Crossings Condition to avoid shell-crossing: R′(t, r)>0. Numerically monitor monotonicity of Riwith r each time step; if violation occurs, refine initial M(r)or t_B(r) profiles to prevent crossings. If unavoidable, implement shock-regularization: merge crossing shells conserving mass and momentum (approximate), but document physically because shell-crossings are unphysical in dust approximation. 4.5 Matching to Exterior FLRW At chosen matching radius (rm), ensure continuity: RLTB(t, rm) = a(t), r(FLRW) m.(3) Enforce M(rm)and E(rm)consistent with exterior curvature: determine E(rm)from (5.10) or direct constraint from extrinsic curvature matching. 5 Curvature Invariant Computation and Quantum Cutoff Compute Kretschmann scalar: K=RαβγδRαβγδ.(10.6) For LTB metric there are closed expressions for K(t, r)in terms of R, ˙ R, R′,¨ Rand M(r). Numerically evaluate Kat each time step and identify first time (tQ) (or radius (RQ)) where K≥KPlanck ≡ℓ−4 P. Use criteria: • Stop classical integration at K= 10−xKPlanck with safety factor (e.g. x= 0 or x= 2) and mark quantum transition. • Record Min enclosed within radius where Kreaches cutoff. 4 6 Numerical Stability, Convergence and Tests 6.1 Convergence Tests •Background integrator: vary integration tolerance and grid resolution for asampling. Verify H(a)converges with expected order (analytical evaluation trivial here). •Root-finding: verify convergence of atunder grid refinement and bracket resolution; confirm root invariance to enlargement of search domain. •LTB integration: perform Richardson extrapolation: run simulations with Nr, Ntdoubled and verify key quantities (collapse time, tAH,RQ) converge with expected order (RK45 is 5th order in time; spatial errors depend on shell discretization but manifest via initial data only). •Constraint monitoring: compute mass conservation (M′(r)) and monitor residuals of Einstein constraints (for numerical LTB, constraints are algebraic but monitor numerical drift). 6.2 Benchmarks and Analytic Limits •Homogeneous limit: set M(r)∝r3,E(r)∝r2to reproduce closed FLRW; compare collapse time to analytic closed-universe result. •Central point mass limit: verify Newtonian potential outside scar matches analytic Φ(r) = −GMs/r at r≫σ. •Perturbation checks: in Boltzmann code, set scar_enable=false and reproduce known ΛCDM spectra. 6.3 Resolution Recommendations •LTB: Nr≥2000 shells for detailed inner-region resolution if σsmall (Mpc scales), otherwise Nr∼500 may suffice. Use nonuniform grid refined near r≲5σ. •Boltzmann runs: keep standard k-grid but add low-krefinement for e f(k;σ)sampling: include log-spaced k points down to kmin ∼10−5Mpc−1if σ≳100 Mpc. 7 Parameter Scan & Likelihood Pipeline 7.1 Grid / MCMC Scanning Strategy •Coarse grid scan: sample log10 |Ωs,0|from (-6) to (-1) (or up to (-0.1)), nin [−2,3] (allows growth and decay), σin [1 Mpc,104Mpc]. Compute at,∆tand discard parameter sets with no future turning (or flag past-turning). •MCMC: for inference, add three new parameters to standard cosmological chain: Omega_s0, n,sigma (and optionally r_s,theta_s,phi_s). Use prior ranges above and include priors enforcing smallness at recombination if desired. •Likelihood evaluation: for each sampled point run modified CLASS to compute CMB Cℓand ISW templates; compute joint likelihood L=LCMB ∗LISW ∗LLSS ∗Lbkg using standard datasets. 5 7.2 Computational Cost and Optimizations • Boltzmann runs are expensive; precompute scar template responses ∂Cℓ/∂Ωs0at fixed (n, σ)and use linear scaling for small amplitudes to accelerate MCMC (surrogate model). • Use emulator / PCA: build a small basis of scar templates over (n, σ)and interpolate using Gaussian process for fast likelihood. 8 Output Products (for Reproducible Paper Figures) Produce the following artifacts for the paper and repository: 1. Tables: •table_a_t.csv listing (Ωs0, n, σ, at,∆tGyr, status). •table_collapse.csv list collapse times ∆tshell(r)for sample parameter sets. 2. Figures: •fig_at_vs_Omega_s0_n.pdf: contour plot of atin (Ωs0, n)plane. •fig_deltaCl.pdf:∆Cℓscar vs baseline for representative parameter choices (SW & ISW decomposition). •fig_LTB_collapse.pdf: shell worldlines R(t, r)showing horizon formation and core. 3. Code snippets: •class_patch.diff and camb_patch.diff with minimal code modifications. •ltb_solver.py or ltb_solver.c with RK integrator, sampling and plotting utilities. 4. Validation notebooks: Jupyter notebooks reproducing unit tests and generating figures. Provide detailed captions and README with units and conventions. 9 Example Default Numerical Setup (Copy-Paste) •H0 = 70 km/s/Mpc,Ωm0= 0.30,ΩΛ0 = 0.70,Ωr0= 9e−5. • Scar fiducial: Ωs0=−1e−3,n=−1.0,σ= 100 Mpc. • CLASS modifications: enable k_min = 1e-5 1/Mpc, k_max = 1.0 1/Mpc, N_k = 200. • LTB: r_max = 5 Gpc (comoving), N_r = 2000 shells refined by r_i = (i/N_r)αwith α= 2 (refinement near center). • Time integrator: RK45 with relative tol 1e-9, absolute tol 1e-12. 10 Reproducibility Checklist 1. Publish input.ini and parameter grid used for all figures. 2. Provide patch diffs for CLASS/CAMB and LTB solver source with exact line numbers (or a git repo). 3. Include unit tests that verify the homogeneous limit, Newtonian limit and SW/ISW smallamplitude checks. 4. Archive raw data for figures and tables and include scripts to regenerate. 6 11 Troubleshooting Common Issues •No root found for (E2(a)=0): expand search range in a, check sign conventions for Ωs0(remember ∆sign mapping), verify nnot producing complex root (take care with fractional exponents when argument negative). •CLASS crash at small k: ensure ftilde=exp(-0.5*(k*�)2)usesdoubleanddoesnotunderflow;implementsafethresholdsif (k*� > 10) ftilde = 0.LTB shell-crossings:refineinitialM(r)orpickmonotonict_B(r);ifmodelingpressurelessdustisinsufficient, switchtohydrodynamicsolver. •• Unphysical negative H² due to rounding near root: clamp to small positive value or implement analytic near-root handling described in 10.2.3. 12 Data Vector and Model Components Consider a dataset composed of: • CMB temperature map (T(ˆn)) (optionally polarization (E(ˆn))). • Galaxy overdensity maps (ga(ˆn)) in redshift bins (a= 1 . . . Nbin). • Lensing convergence map (κ(ˆn)) (optional). We model the observed CMB harmonic coefficients as aobs ℓm =aprim ℓm +A, tℓm(θscar) + nℓm,(11.1) where •aprim ℓm are primordial CMB multipoles from standard ΛCDM fluctuations (Gaussian with known theoretical covariances), •tℓm(θscar)is a deterministic scar template normalized to unit amplitude and dependent on scar parameters θscar ={σ, n, rs,ˆns}, •Ais the scalar amplitude parameter proportional to Ωs,0(linear scaling for small amplitude), •nℓm is instrument noise (assumed Gaussian with known noise power spectrum Nℓ). We will treat aprim ℓm as a Gaussian random field with covariance ⟨aprim ℓm aprim∗ ℓ′m′⟩=CΛCDM ℓδℓℓ′δmm′. Masking and beam effects are handled below. 13 Matched-Filter (Optimal Linear Estimator) — Derivation When searching for a small deterministic template (tℓm) in a Gaussian background with covariance C, the optimal linear unbiased estimator for amplitude (A) is the matched filter (generalized least squares). In harmonic space define vectorized data acontaining all aobs ℓm up to ℓmax. The log-likelihood (Gaussian) is −2 ln L(A) = (a−At)†C−1(a−At) + const.(11.2) Maximizing over (A) gives ˆ A=t†C−1a t†C−1t.(11.3) Variance of ˆ A(Fisher inversion) is Var[ ˆ A] = σ2 A=t†C−1t−1.(11.4) 7 Explicit harmonic form (full-sky, diagonal covariance) with Ctot ℓ=CΛCDM ℓ+Nℓ: ˆ A=Pℓm t∗ ℓm, aobs ℓm /Ctot ℓ Pℓm |tℓm|2/Ctot ℓ , σ−2 A=X ℓm |tℓm|2 Ctot ℓ .(11.5) When the template depends on extra parameters (position ˆnsor σ) you can either: (i) evaluate ˆ Aon a grid of template parameter values and pick maximum-likelihood point; or (ii) include template derivatives and perform a joint Fisher/MCMC to explore parameter space (see Sec. 11.6). 14 Masking, Partial-Sky and Pseudo-CℓCorrections Real maps use a mask M(ˆn)(0/1 or apodized) which couples harmonic modes. Let masked spherical harmonic coefficients be ˜aℓm =Zdˆn M(ˆn), T(ˆn), Y ∗ ℓm(ˆn).(11.6) The expectation of the pseudo power ˜ Cℓ=1 2ℓ+1 Pm|˜aℓm|2is related to the true power by mode-coupling matrix (Mℓℓ′): ⟨˜ Cℓ⟩=X ℓ′ Mℓℓ′, Cℓ′.(11.7) For template fitting, the correct matched-filter on the cut sky replaces Cby the full cut-sky covariance ˜ C(non-diagonal). In practice two equivalent and computationally efficient strategies are common: 14.1 Strategy 1 — Inverse-Variance Filtering in Pixel Space 1. Build pixel-space covariance Cpp′including noise and beam for masked maps (set infinite variance for masked pixels or remove them by reducing vector). 2. Compute filtered map x=C−1dusing conjugate-gradient solvers (preconditioned). 3. Evaluate ˆ A= (t⊤x)/(t⊤C−1t)with tin pixel space (template projected to pixel space and masked). This is exact for Gaussian fields and optimal. 14.2 Strategy 2 — Pseudo-Harmonic Matched Filter with Mode-Coupling Correction Compute masked-template harmonics ˜ tℓm and masked data ˜aℓm. Use approximate inverse covariance ˜ C−1 ℓ(diagonal approximation) and correct estimator bias via coupling matrix: ˆ A≈Pℓm ˜ t∗ ℓm˜aℓm/˜ Cℓ Pℓm |˜ tℓm|2/˜ Cℓ ,(11.8) then debias normalization by Monte Carlo: compute ⟨ˆ A⟩on many Gaussian ΛCDM realizations (with mask/beam/noise) and apply multiplicative correction (b) such that ˆ Adebias =ˆ A/b. This approach is faster but approximate; use it for wide parameter scans and confirm promising candidates with Strategy 1. 8 Mode-coupling matrix (Mℓℓ′): compute from mask spherical harmonics (wLM ): Mℓℓ′=2ℓ′+ 1 4πX L (2L+ 1)wLℓ ℓ′L 0 0 02 ,(11.9) with wLthe power spectrum of the mask. Use this to compute expected variance increase due to mask for significance estimates. 15 Cross-Correlation Estimator: ISW–Galaxy Matched Estimator The scar produces an ISW signal that correlates with large-scale structure. For a galaxy map (ga(ˆn)) in bin (a) with spherical harmonic coefficients ga ℓm, the cross-spectrum estimator is ˆ CTga ℓ=1 2ℓ+ 1 X m aobs ℓm , ga∗ ℓm.(11.10) Its expectation in presence of a scar is ⟨ˆ CTga ℓ⟩=CTga,,ΛCDM ℓ+A, Ctga ℓ,(11.11) where Ctga ℓis the cross-spectrum between the scar template and galaxy bin (a) (computable given bias and selection function). For significance, use band-averaged cross-spectra or multipole range ℓ≲50 to focus on ISW scales. Covariance between ˆ CTg ℓestimators (Gaussian approximation): Covˆ CTga ℓ,ˆ CTgb ℓ′=δℓℓ′ (2ℓ+ 1)fsky hCT T ℓ+NTT ℓCgagb ℓ+Ngagb ℓ+CTga ℓCTgb ℓi,(11.12) where fsky is sky fraction after mask. Use this covariance in joint likelihoods combining CMB and LSS. Optimal estimator for Afrom cross-correlation: Form weighted sum across multipoles and bins: ˆ ATg =Pℓ,a,b(Ctga ℓ)∗[Cov−1]ℓa,ℓb ˆ C∗ℓTgb Pℓ,a,b(Ctga ℓ)∗[Cov−1]ℓa,ℓb C∗ℓtgb.(11.13) This generalizes the matched filter to cross correlation. 16 Joint Likelihood and Marginalization over Cosmological Parameters A robust inference must marginalize over primary cosmological parameters ϕ(e.g., Ωbh2,Ωch2, H0, ns, As, τ). Let model prediction for harmonic covariance be C(ϕ) + A, T(θscar)where Tis the scar contribution to covariance (or to mean if treated as deterministic). The full posterior is P(A, θscar,ϕ|a)∝ L(a|A, θscar,ϕ), π(A), π(θscar), π(ϕ).(11.14) Two practical options: 9 16.1 Option 1 — Profile Likelihood (Fast) 1. For each scar parameter grid point ((A, θscar)) compute χ2(ϕ)and maximize over ϕ(using existing fast Boltzmann solvers for the cosmology). 2. Use ∆χ2to get frequentist confidence intervals on (A). This is faster but may underestimate uncertainties if strong degeneracies exist. 16.2 Option 2 — Full MCMC Marginalization (Recommended for Final Constraints) 1. Add scar parameters to MCMC sampler (MontePython / CosmoMC) and run joint chains using modified CLASS (scar enabled). 2. Use priors on A(e.g., Gaussian or flat) and on θscar. 3. Produce marginalized posterior (P(A)) and 2D contours with other parameters. Computational tip: because full CLASS runs per MCMC step are expensive, use surrogate modeling: precompute derivatives ∂Cℓ/∂A and treat scar contribution linear in (A) for small amplitude. For θscar (e.g., σ), build an emulator grid and interpolate. 17 Fisher Forecasting for Scar Parameters (Derivation) When planning surveys or reporting projected limits, the Fisher matrix (for Gaussian likelihood) is Fij =1 2X ℓ (2ℓ+ 1)fsky,TrC−1 ℓ ∂Cℓ ∂θi C−1 ℓ ∂Cℓ ∂θj.(11.15) If scar contributes only to mean (not covariance) and is small, a simplified Fisher for amplitude (A) is σ−2 A≈X ℓm |tℓm|2 Ctot ℓ≈X ℓ 2ℓ+ 1 Ctot ℓ ˆ t2 ℓ,(11.16) with ˆ tℓ=1 2ℓ+1 Pm|tℓm|2(template power per ℓ). This matches matched-filter variance (11.5). Use this to compute forecast σ(Ωs,0)by mapping A→Ωs,0via linear scaling (A=α, Ωs,0) (compute αby running one full model evaluation). 18 Null Tests, Monte Carlo Calibration and Look-Elsewhere Because low-ℓstatistics are cosmic-variance dominated and susceptible to a posteriori selection effects, rigorous analysis requires: 1. Null simulations: generate NGaussian ΛCDM mock maps including noise and mask; compute ˆ Afor each to build null distribution (P0(ˆ A)). 2. Calibration of biases: compute mean ⟨ˆ A⟩null and subtract observed ˆ Abias if present; compute empirical σA. 3. Look-elsewhere correction: if scanning over position ˆns, width σ, and exponent (n), derive empirical p-value by counting fraction of null realizations where maxθˆ Anull(θ)≥ ˆ Amax data. This controls false-positive rate. 4. Cross-validation: split data (e.g., frequency channels, half-mission maps) to confirm signal stability and to check for foreground residuals. 10 19 Treatment of Foregrounds and Systematics Low multipoles are susceptible to Galactic foregrounds and instrumental systematics. Recommended steps: • Use multiple component-separation CMB maps (SMICA, SEVEM, Commander); require signal stability across maps. • Mask conservative Galactic region (e.g., fsky ≥0.6usable) and apodize edges to reduce mode coupling. • Check frequency dependence: a cosmological scar is achromatic in thermodynamic temperature units; any frequency dependence signals foreground residual. • Null tests against templates of known systematics (scan-strategy stripes, map-making residuals). • For ISW–galaxy analyses control for survey systematics (seeing, star contamination) by applying weights and null corrections. 20 Converting Amplitude Constraints to Ωs,0and Physical Interpretation Estimate conversion factor by linearizing the Boltzmann response: ∆Cℓ≈∂Cℓ ∂Ωs,0Ωs,0=0∆Ωs,0,(11.17) so if your matched-filter is normalized to unit Ωs,0then (A) equals Ωs,0. In practice compute numerically: 1. Run CLASS with Ωs,0= +δand −δ(small) to compute ∂Cℓ/∂Ωs,0≈∆Cℓ/(2δ). 2. Compute template harmonics (tℓm) corresponding to Ωs,0=δ, then matched-filter outputs ˆ Ahave units of Ωs,0. Thus upper limits on (A) translate directly into bounds on Ωs,0. Report bounds like Ωs,0< X (95% CL), and convert to constraints on ator turn time using analytic relations from Sec. 4. 21 Worked Example — Recipe for a Real Data Analysis (CopyPaste Ready) 1. Data preparation • Download Planck SMICA temperature map (HEALPix Nside = 2048), corresponding mask (e.g., common confidence mask). • Smooth to 1◦if desired; degrade to Nside = 512 for low-ℓanalysis. 2. Template generation • For grid of {σ, n, rs}generate scar potential Φs(x, η)using Poisson approximation and project to temperature template t(ˆn)including SW+ISW integrals (Sec. 7 formulas). • Convert to spherical harmonics tℓm up to ℓmax = 60. 11 3. Matched-filtering • Option A (fast): compute masked ˜ tℓm,˜aℓm, evaluate Eq. (11.8), debias using 2000 ΛCDM null sims. • Option B (accurate): build pixel-space covariance and perform inverse-variance filtering (recommended for final result). 4. Null calibration and significance • Generate N= 2000 simulations with same beam, noise, mask; compute ˆ Anull grid distribution; derive empirical p-value for observed ˆ Aand apply look-elsewhere correction across parameter grid. 5. Cross-checks • Repeat for Commander, SEVEM maps; check frequency maps; test half-mission splits. • Compute cross-correlation with NVSS galaxy map using Eq. (11.10); check direction alignment with best-fit scar. 6. Reporting • If no detection: report Ωs,0<Ωlim at 95% CL, show posterior and 2D constraints with (n, σ). • If detection: show stability tests, foreground nulls, and joint likelihood with LSS. 22 Present Constraints & Recommended Priors for MCMC For a conservative blind analysis choose priors: •Ωs,0uniform in log between [−10−6,−10−1]if expecting negative sign, or symmetric around zero if sign unknown. •nuniform on [−2,3]. •log10 σ/Mpc uniform over [0,4] (1 Mpc—10 kpc to 10 Gpc); refine after preliminary runs. For forecasts include stronger prior on position (rs) (if searching for off-center signals) or fix to centered model to reduce parameter space. 23 Caveats and Failure Modes •Degeneracy with cosmic variance: low-ℓis sample-variance limited. Detection of small |Ωs,0|near cosmic-variance floor may be ambiguous—require multiple corroborating channels (ISW–galaxy, lensing). •Foreground leakage: residual Galactic foregrounds can mimic large-scale anisotropy; rigorous frequency tests and component-separation stability required. •Model dependence: mapping ˆ A→atdepends on assumed nand averaging fraction; report results for multiple plausible mappings. 12 24 Summary (Operational Checklist for the Paper) 1. Define template family tℓm(θscar); produce public repository with templates and code to compute them. 2. Use matched filter (11.3) with inverse-variance filtering and rigorous Monte-Carlo null calibration. 3. Jointly analyze CMB + ISW–galaxy cross correlation (11.13) to lift degeneracies. 4. Marginalize cosmological parameters via MCMC or profile likelihood; report robust upper limits or detections with look-elsewhere correction. 5. Provide reproducible code, null-simulation set, and scripts to convert amplitude to Ωs,0 and to turning-time (at). 25 Effective Variational Origin (Operational Ansatz) We start with the Einstein–Hilbert action plus a phenomenological scar action (Ss). Work in geometric units (c= 1) until otherwise noted. Stot =1 16πG Zd4x, √−g, (R−2Λ) + Sm[gµν,Ψ] + Ss[gµν; Θ],(12.1) where Smis the matter action (standard fluids/fields Ψ) and Ssis an effective action encoding the scar microphysics and depends on a finite set of phenomenological parameters Θ(e.g. amplitude, width σ, EoS parameter ws, localization center xs). Define the scar stress–energy tensor by functional variation as usual: ∆µν ≡ − 2 √−g δSs δgµν .(12.2) Then the field equations read Gµν + Λgµν = 8πGTµν +T(scar) µν , T(scar) µν ≡ − 1 8πG∆µν.(12.3) Assume Ssis diffeomorphism invariant (or constructed such that ∇µ∆µν = 0 holds); this ensures Bianchi consistency without energy exchange terms (see discussion in main text). Next we specify a concrete local ansatz. 26 Scar Ansatz (Gaussian Localized Perfect-Fluid Form) — Explicit Components Adopt comoving FLRW coordinates with signature (−+ ++): ds2=−dt2+a2(t)γijdxidxj, γijdxidxj=dr2 1−kr2+r2dΩ2.(12.4) Phenomenological ansatz for ∆µν (Section 3 style): ∆µν(t, x) = 8πG(ρs(t) + ps(t))uµuν+ ps(t)gµνfG(r;σ); where fG(r;σ) = 1 (2π)3/2σ3exp−r2 2σ2, r ≡ |x−xs|.(12.6) Take comoving observers (uµ= (1,0,0,0)), (uµ=gµνuν= (−1,0,0,0)). Compute components explicitly. 13 for a condensed core where gradients inside the core are small. Then (∇iχ≈0) inside the core and (∇0χ= ˙χ0(t)). Under this assumption the stress tensor (B.5) simplifies inside the core: •Time-time component: ∆00 =F(r;σ)(˙χ2 0−1 2(−1) ˙χ2 0−(−1)V(χ0))=F, (1 2˙χ2 0+V(χ0)).(B.9) •Spatial components: ∆ij =F(r;σ); a2(t), δij;(1 2˙χ2 0−V(χ0)).(B.10) Therefore define local effective energy density and pressure (as they appear on the geometric side through (∆µν)): ρ(local) s(t, r) := 1 2˙χ2 0+V(χ0)(multiplied by F(r;σ)in ∆00),(B.11) p(local) s(t, r) := 1 2˙χ2 0−V(χ0)(appearing with factor F(r;σ)).(B.12) Remember: depending on your placement (left-hand vs right-hand side) these appear with a sign flip when defining (T(scar) µν ). Be consistent with the sign convention used in the main text. To obtain the volume-averaged quantities used in the Friedmann equation, integrate against the kernel over a large domain (D) containing the scar: hρsiD(t)≡1 VD∫D d3x ρ(local) s(t, r), F (r;σ).(B.13) If (F) is normalized so (∫d3x, F = 1) and the domain (D) is large compared to the kernel, this reduces approximately to ρs,avg(t)≃ρ(core) s(t)×fV;, fV≡Vcore VD1,(B.14) where (ρ(core) s(t)≡1 2˙χ2 0+V(χ0)) and (fV) is the small volume fraction the scar occupies in the averaging domain. In cosmological notation define (ρs,0) at (a= 1) by (ρs,0≡ρs,avg(a= 1)), and the normalized density (Ωs,0= 8πGρs,0/(3H2 0)). This matches the phenomenological parameter used in Sections 3–12. 5 Mapping to phenomenological scaling (ρs(a) = ρs,0a−n) We can generate an effective scaling exponent (n) from the dynamics of (χ0(t)). Two simple regimes: 5.1 (i) Coherent oscillations (massive scalar, (mH)) If (V(χ) = 1 2m2χ2) and the field oscillates rapidly compared to Hubble ((mH)), then the averaged equation of state for a quadratic potential is (hwi ≃ 0) and the energy density of the oscillating condensate scales like matter: ρ(core) s(t)∝a−3(t) =⇒n≃3.(B.15) 3 5.2 (ii) Slow-roll or potential-dominated regime (like vacuum energy) If (χ) is slow-rolling and potential dominated (( ˙χ2V), (V) approximately constant) then (ρ(core) s≃V(χ0)) nearly constant and ρs,avg(a)≈const ×fV=⇒n≃0.(B.16) 5.3 (iii) Exotic possibilities (growing with (a)) If the condensate couples to background expansion or to curvature in a time-dependent way (or if (F) is taken to be a function of proper radius rather than comoving radius), one may obtain effective (n < 0) (i.e., (ρs) that grows with (a)). For instance, if the physical core radius is fixed (size in physical coordinates), then a comoving kernel shrinks effectively like (a−3) in volume fraction and the averaged (ρs,avg) could scale with positive powers of (a). Several microphysical routes to growing (ρs) exist (e.g., accretion of ambient matter into the core during contraction, or a condensate whose amplitude grows due to couplings); those require specifying interactions beyond the minimal action (B.1). The toy action above gives explicit control to realize specific (n) by choice of potential and initial conditions. 6 Covariant conservation (∇µ∆µν = 0) and consistency If (Ss) is diffeomorphism invariant and (F) is a scalar (time-independent comoving scalar), variation with respect to diffeomorphisms yields the covariant conservation identity for the stress tensor derived from (Ss). Explicit check: Using the field equation (B.8) and the identity for scalar-field stress tensor divergence one finds ∇µ∆µν =F(□χ−V′(χ)),∇νχ+ (∇νF),(1 2∇αχ∇αχ+V(χ)).(B.17) Upon imposing the scalar EOM (B.7) this reduces to ∇µ∆µν = (∇νF),(1 2∇αχ∇αχ+V(χ))−∇µF F∇µχ, ∇νχ. (B.18) If (F) is strictly time-independent in comoving slicing and (χ) is homogeneous in the core (so (∇iχ≈0) and (∇iF) nonzero only near edges), the right-hand side is suppressed (localized at the rim). In the averaging procedure over large domain (D) these surface/localized terms become negligible for global dynamics and one obtains to very good approximation (h∇µ∆µνiD≃0). Operationally this justifies using (∆µν) as a covariantly conserved effective contribution in the Friedmann equation (Sections 3–12). If one desires exact conservation, promote (F) to a covariantly-constructed scalar (e.g., function of spatial geodesic distance), in which case the EOM (B.7) ensures (∇µ∆µν = 0) identically. For the practical phenomenology in the paper the approximate conservation above suffices. 7 Small-gradient correction and edge effects (explicit expression) For completeness we give the full form (no approximations) of (∆µν) when (χ=χ(t, r)) varies radially and (F=F(r)). Using (B.5) with explicit metric: ∆tt =F(1 2˙χ2+1 2a2χ′2+V),(B.19) 4 ∆tr =F, ˙χ, χ′,(B.20) ∆rr =F(χ′2−1 2grr(˙χ2−1 a2χ′2)−grrV),(B.21) and angular components similar. Here prime (’) is (∂r). These formulae allow one to compute edge-layer contributions exactly in LTB or spherical coordinates and to evaluate (∇µ∆µν) explicitly if needed for junction matching. They are ready to implement numerically. 8 Practical parameter mapping and prescription to paste into the paper Include the following boxed recipe in the manuscript so readers can map model parameters: Box (Parameter mapping — copy-paste): Starting from action (B.1) with Gaussian kernel (F(r;σ)) normalized so (∫d3x, F = 1), solve the scalar EOM (B.8) for (χ0(t)) under the homogeneous-core ansatz. Then define ρs,core(t) = 1 2˙χ2 0+V(χ0), ps,core(t) = 1 2˙χ2 0−V(χ0).(3) The averaged scar density entering the Friedmann equation is ρs,avg(t)≃fV, ρs,core(t),(4) with (fV≡(2π)3/2σ3 VD ) the core volume fraction for domain (D). Define (Ωs,0= 8πGρs,avg(a= 1)/(3H2 0)). Choose potential (V(χ)) and initial (χ0(tinit)) to engineer desired effective exponent (n) via (ρs,avg(a)∝a−n) (e.g. (n≈3) for oscillating massive field; (n≈0) for slow-roll). This boxed text is ready to paste in Methods/Appendix and links the microphysical action to the phenomenological parameters (Ωs,0) and (n) used elsewhere in the paper. 9 Ready-to-paste LaTeX snippet Below is a compact LaTeX-ready block that you can paste into Appendix B of your manuscript; it reproduces the main derivations above (wrap in \begin{appendix} if needed): \section*{Appendix B: Microphysical realization of the scar — localized scalar condensate} We consider the effective action \[ S_s[\chi,g] = -\int d^4x\;\sqrt{-g}\;\Big[\tfrac12 g^{\mu\nu}\nabla_\mu\chi\nabla_\nu\chi + V(\chi)\Big]\;F(r;\sigma), \] with $F(r;\sigma)$ a fixed comoving kernel (e.g. Gaussian $F=(2\pi)^{-3/2}\sigma^{-3}e^{-r^2/(2\sigma^2)}$). Variation with respect to $g^{\mu\nu}$ yields \[ \Delta_{\mu\nu} \equiv -\frac{2}{\sqrt{-g}}\frac{\delta S_s}{\delta g^{\mu\nu}} = F(r;\sigma)\Big(\nabla_\mu\chi\nabla_\nu\chi - \tfrac12 g_{\mu\nu}\nabla^\alpha\chi\nabla_\alpha\chi - g_{\mu\nu}V(\chi)\Big), \] which is the explicit residual tensor appearing in the modified Einstein equations used in the main text. Variation with respect to $\chi$ gives the equation of motion \[ \Box\chi - V'(\chi) + \frac{\nabla^\mu F}{F}\nabla_\mu\chi = 0. \] 5 Assuming a homogeneous core $\chi(t,\mathbf x)\simeq\chi_0(t)$ for $r\lesssim\sigma$, the local energy density and pressure are \[ \rho_{s}^{\rm core}(t)=\tfrac12\dot\chi_0^2 + V(\chi_0),\qquad p_{s}^{\rm core}(t)=\tfrac12\dot\chi_0^2 - V(\chi_0), \] and the volume--averaged scar density entering the Friedmann equation is $\rho_{s,\rm avg}(t)\simeq f_V\,\rho_{s}^{\rm core}(t)$ where $f_V$ is the core's comoving volume fraction. Choosing $V(\chi)$ and initial data for $\chi_0$ allows one to engineer effective scalings $\rho_{s,\rm avg}\propto a^{-n}$ (e.g.~$n\approx3$ for an oscillating massive scalar, $n\approx0$ for slow-roll). 10 Final remarks This Appendix provides a minimal, explicit and controllable microphysical model that yields the phenomenological residual tensor (∆µν) used throughout the paper. It is simple enough to be implemented numerically, general enough to generate desired scaling behaviours by choice of potential and kernel, and flexible enough to be extended (multiple fields, internal degrees of freedom, covariant kernel) when constructing a fuller UV completion. 6 11 Notation recap We use the LTB metric and standard functions: ds2=−dt2+(R′(t, r))2 1+2E(r)dr2+R2(t, r)dΩ2,(5) ˙ R2(t, r) = 2M(r) R(t, r)+ 2E(r), M′(r) = 4πρ(t, r)R2R′.(6) Primes denote (∂r), dots denote (∂t). The Misner–Sharp mass (M(r)) is the (time-independent for dust) enclosed mass inside comoving label (r). The energy function (E(r)) controls bound/marginal/unbound shell dynamics. We adopt initial data at the turning time (t=tt): R(tt, r) = at, r, ˙ R(tt, r) = 0.(7) From the latter and the evolution equation one has the constraint at (tt): 0 = 2M(r) atr+ 2E(r) =⇒E(r) = −M(r) atr.(14.0) This relation will be used in several derivations below. 12 Case A — Marginally bound shells: (E(r)=0) (exact solution) When (E(r) = 0) the shell equation reduces to ˙ R2=2M(r) R.(8) Choose the collapsing branch (negative ( ˙ R) after turning). Integrate: ˙ R=−√2M R⇒∫R1/2dR =−√2M∫dt. (9) Perform the integral: 2 3R3/2=−√2M, (t−tB(r)) (10) so the general solution is R(t, r) = [3 2√2M(r),(tB(r)−t)]2/3;.(14.1) Here (tB(r)) is the (shell-dependent) bang/collapse time. Using initial condition at turning time (tt) with (R(tt, r) = atr), solve for (tB(r)): tB(r) = tt+2 3 (atr)3/2 √2M(r).(14.2) Collapse time from turning to singularity for shell (r) is then ∆tcoll(r)≡tB(r)−tt=2 3 (atr)3/2 √2M(r)=1 3√2,a3/2 t, r3/2 √M(r);.(14.3) This is exact for (E= 0). Note numerical prefactor differs from elliptic case constants; keep numerical factors exact. 7 13 Case B — Elliptic shells: (E(r)<0) (parametric solution, closed form) Set (E(r) = −1 2k(r)) with (k(r)>0). The evolution equation is ˙ R2=2M R−k. (11) Parametric solution (standard, analogous to closed FLRW) uses angle parameter (η): R(t, r) = 2M(r) k(r)(1−cos η), t−tB(r) = 2M(r) k3/2(r)(η−sin η), (14.4) where (0< η ≤2π). Collapse to (R= 0) corresponds to (η= 2π). Using turning-time initial condition: at turning ( ˙ R(tt, r) = 0) which corresponds to (η=π); at (η=π) we have R(tt, r) = 2M k,(1 −cos π) = 4M k.(12) But (R(tt, r) = atr), so k(r) = 4M(r) atr.(14.5) Insert (14.5) into parametric formulas to obtain explicit collapse time. Time from turning ((η=π)) to collapse ((η= 2π)) ∆tcoll(r) = t(η= 2π)−t(η=π) = 2M k3/2[(2π−0) −(π−0)]=2M k3/2π. (13) Substitute (k= 4M/(atr)): ∆tcoll(r) = 2M (4M atr)3/2π= 2M(a3/2 tr3/2 (4M)3/2)π =π 21/2,a3/2 tr3/2 M1/2·1 2=π 23/2,a3/2 tr3/2 √M(r). (14.6) Thus the elliptic collapse time (turning→singularity) is ∆tcoll(r) = π 23/2 a3/2 tr3/2 √M(r);.(14.7) Compare with marginally bound (14.3); the numerical factor difference is ( π 23/2/1 3√2= 3π 4≈2.356) — elliptic shells collapse sooner (factor dependent) than marginally bound with same M(r)and initial R(tt). Parametric (η(t)) inversion for elliptic case useful when needing explicit (t7→ R) relation: from (14.4) η−sin η=k3/2 2M(t−tB).(14) One can solve numerically for (η) given (t) if needed. 8 14 Case C — Gaussian scar mass profile: explicit (M(r)) and small/large-rasymptotics Adopt a physically reasonable, smooth central scar mass profile (M(r)) which transitions from a central mass (Ms) inside core to standard environmental mass at large (r). A convenient, analytically integrable choice is the cumulative of a Gaussian density kernel: M(r) = Ms;FM(r;σ) + 4π 3ρenv, r3,(14.8) with FM(r;σ)≡erf (r √2σ)−√2 π r σe−r2/(2σ2),(15) scaled so that (M(r→ ∞)→Ms+ (4π/3)ρenvr3) (the exact prefactor for normalization can be adjusted — here (FM(r;σ)) is chosen to give finite central mass (Ms) at (r→ ∞) when integrated against normalized kernel). An alternative compact form is (useful and often simpler) M(r) = Ms(1−e−r2/(2σ2))+4π 3ρenv, r3,(14.9) which captures the same qualitative behavior (central mass (Ms) inside (r≲σ) and cubic growth at large (r)). We will use (14.9) for algebraic simplicity in closed-form approximations. Exact small-rexpansion (for (14.9)): For (rσ), M(r)≈Ms(1−(1−r2 2σ2+r4 8σ4+···))+4π 3ρenvr3=Ms r2 2σ2+4π 3ρenvr3+O(r4).(14.10) Hence near the center (M(r)∝r2) (if using the simple cut form (14.9)), while a more physically normalized Gaussian cumulative would give (M(r)∝r3) near the origin. Use whichever normalization is intended — ensure regularity (M(r)∼r3) as (r→0) to avoid central mass singularities. If exact regularity is required, choose M(r) = Ms[1−exp (−r3 r3 0)]+4π 3ρenvr3,(16) or use a proper normalized density (ρs(r) = ρs,0, e−r2/2σ2) so that (M(r) = 4πρs,0∫r 0˜r2e−˜r2/2σ2d˜r∝ r3) near zero. Large-rbehavior: For (rσ), (e−r2/2σ2→0) and M(r)∼Ms+4π 3ρenvr3.(14.11) Therefore inner shells see effective mass dominated by (Ms) while outer shells see growing cubic environmental mass. 15 Explicit collapse time for Gaussian-type (M(r)): substitution into (14.7) Using the elliptic collapse-time formula (14.7) and substituting (M(r)) from (14.9) (or the physically normalized variant), we obtain 9 ∆tcoll(r) := π 23/2,a3/2 tr3/2 √Ms(1−e−r2/(2σ2))+4π 3ρenvr3 ;.(14.12) This is an explicit closed-form expression (elementary functions only) suitable for plotting or asymptotic analysis. Key asymptotic regimes: 1. Inner-core regime (rrc) where (M(r)≈βr3) (homogeneous-like): if (M(r)≈ (4π/3)ρeffr3), ∆tcoll(r)≈π 23/2,a3/2 tr3/2 √(4π/3)ρeffr3=π 23/2,a3/2 t √(4π/3)ρeff , r0,(17) so collapse time becomes independent of (r)— homogeneous simultaneous collapse (synchronous crunch). 2. Scar-dominated inner regime (M(r)≈Ms) (for (r) small but (Ms) constant dominant): ∆tcoll(r)≈π 23/2,a3/2 tr3/2 √Ms∝r3/2.(18) So inner shells collapse much faster (funnelling) — small (r)→small (∆t). 3. Outer regime where environmental (r3) dominates: then similar to (1) homogeneous scaling giving (r)-independent collapse time for those shells. These formulae let you quantify funneling: plot (∆tcoll(r)) vs (r) for given (Ms, σ, ρenv) to show inner-shell rapid collapse. 16 Apparent horizon condition: explicit evaluation Apparent horizon in LTB occurs when R(t, r) = 2M(r).(14.13) Using parametric solution (14.4) for elliptic shells: 2M k(1 −cos ηAH) = 2M=⇒1−cos ηAH =k, (19) but recall (k) is defined so that (2E=−k) and from (14.5) (k= 4M/(atr)). Thus 1−cos ηAH =4M atr.(14.14) Feasible solution requires RHS ≤2(since (1−cos η≤2)), which yields a geometric constraint on radii and parameters: shells satisfying (4M/(atr)≤2) or (2M≤atr). This is consistent: at turning (R(tt, r) = atr), and apparent-horizon condition (R= 2M) is satisfied at turning only if (atr= 2M(r)) — shells that were already trapped at turning. Time of AH crossing: Solve parametric eqn for (ηAH) by cos ηAH = 1 −4M atr.(20) 10 Then time of crossing (tAH) is tAH −tB=2M k3/2(ηAH −sin ηAH) = 2M (4M atr)3/2(ηAH −sin ηAH).(14.15) Using substitution this is explicit but involves (ηAH) which is given above in closed form by inverse cosine. Thus (tAH) is explicit (elementary) and ready to evaluate numerically or expand asymptotically. 17 Small-rfunneling asymptotics (useful inequality) Assume inner mass behaves (M(r)∝rα) as (r→0). Then from elliptic formula (14.7) ∆tcoll(r)∝r3/2 √M(r)∝r3/2−α/2.(21) • If (α < 3) (mass grows slower than (r3)), then exponent (3/2−α/2>0) and (∆t→0) as (r→0): strong funneling (inner shells collapse faster). • If (α= 3) (homogeneous), exponent zero: simultaneous collapse. • If (α > 3) (unphysical for regular core), inner shells collapse slower. Thus a sufficient condition for funneling is α < 3 =⇒funnelling (inner shells collapse earlier).(14.16) For practical Gaussian-like cores choose parameters so that effective (α≲2) in inner region to get strong funneling. 18 Example numeric-ready expressions (copy-paste) For convenience, here are compact expressions you can paste directly into the manuscript or code comments: • Elliptic collapse time: ∆tcoll(r) = πa3/2 t 23/2;r3/2 √M(r).(22) • Marginal case collapse time: ∆t(E=0) coll (r) = a3/2 t 3√2;r3/2 √M(r).(23) • Gaussian-cut mass model (practical): M(r) = Ms(1−e−r2/2σ2)+4π 3ρenvr3.(24) • Substitution yields ∆tcoll(r) = πa3/2 tr3/2 23/2√Ms(1−e−r2/2σ2)+4π 3ρenvr3 .(25) 11 • Apparent horizon crossing condition (explicit): ηAH = arccos (1−4M(r) atr), tAH =tB+2M(r) (4M(r) atr)3/2(ηAH −sin ηAH).(26) These are directly evaluable numerically and suitable for inclusion in Figures/Tables. 19 Practical implementation notes 1. Choice of (M(r)) regularity: ensure (M(r)∼r3) near (r→0) for smooth regular core (avoid conical singularity). Use a normalized Gaussian density (ρs(r) = ρs,0e−r2/2σ2) so (M(r) = 4πρs,0∫r 0˜r2e−˜r2/2σ2d˜r). 2. Handling small denominators: when (M(r)) is very small, collapse times can become numerically large; clamp numerical evaluation using analytic small-rexpansions where appropriate. 3. Comparing elliptic vs marginal: use (14.3) and (14.7) interchangeably as sanity checks; in transitional regions where (E(r)≈0) both formulas should be consistent within numeric prefactors. 4. Units: if restoring (G) and (c), insert (√2GM) etc.; collapse times scale as ((a3/2 tr3/2)/√GM) in SI. 12